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EDL Sixth Year Theses Isabelle Farrington College Of Education

Fall 2015

The Impact of Early Numeracy Intervention onKindergarten StudentsJennifer L. HillSacred Heart University, [email protected]

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THE IMPACT OF EARLY NUMERACY INTERVENTION 1

The Impact of Early Numeracy Intervention

on Kindergarten Students

Jennifer Hill

Sacred Heart University

Advisor: Michael K. Barbour


Abstract

The purpose of this study was to explore the effectiveness of early numeracy intervention with

kindergarten students. In order to grow a stronger understanding of how providing mathematics

intervention can benefit students, the intervention was provided to kindergarten students aimed

to seek answers to how providing the earliest possible intervention can positively impact the

achievement gap and a child’s understanding of number. This study explored the impact early

numeracy intervention had on five kindergarten students and compared their growth to those of

their peers not receiving intervention to determine the positive impact providing Response to

Intervention had on kindergarten students. Quantitative and qualitative methods of data were

collected using the AIMSweb Test of Early Numeracy, observations, and student work samples

were collected to analyze the effectiveness of the intervention. Independent sample t-tests

alongside of the constant comparative methods of data analyses were done to triangulate all

sources of data all indicated that the early numeracy intervention had a positive impact on the

five students whom participated. Over a six week Tier 2 intervention cycle, each of the five

students improved their ability to count, understand cardinality, discriminate between numbers

one through ten, and began improving their knowledge of the counting sequence within ten.


Table of Contents

Abstract ……………………………………………………………………………………….. 2

Table of Contents ……………………………………………………………………………... 3

Chapter 1: Introduction ……………………………………………………………………… 5

A. Introduction………………………………………………………………….. 5

B. Statement of the Problem …………………………………………………… 6

C. Thesis Study ………………………………………………………………… 6

D. Summary ……………………………………………………………………. 8

Chapter 2: Literature Review …………………………………………………………………9

A. Introduction …………………………………………………………………..9

B. Number Sense ………………………………………………………………. 11

C. Base 10 ……………………………………………………………………… 17

D. Hands-on Approach ………………………………………………………… 21

E. Response to Intervention ……………………………………………………. 24

F. Summary …………………………………………………………………….. 33

Chapter 3: Research Purpose and Questions ……………………………………………….. 35

A. Research Questions …………………………………………………………..35

B. Methodology ………………………………………………………………... 35

C. The Case …………………………………………………………………….. 37

D. Data Collection Methods …………………………………………………….38

E. Data Analysis Methods ………………………………………………..……. 43

F. Reliability and Validity ……………………………………………………… 47

E. Summary …………………………………………………………………….. 49


Chapter 4: Results and Discussion …………………………………………………………... 51

A. Overview of Case Study ……………………………………………………..51

B. Results and Discussion to Research Question One………………………….. 53

C. Results and Discussion to Research Question Two ………………………… 61

D. Summary ……………………………………………………………………. 67

Chapter 5: Conclusion …………………………………………………………………………68

A. Summary of Case Study …………………………………………………….. 68

B. Implications of Practice ………………………………………………………71

C. Suggestions for Future Research ……………………………………………..73

D. Limitations ………………………………………………………………...…74

References ………………………………………………………………………………………76

Appendices ……………………………………………………………………………………...83

A. AIMSweb Test of Early Numeracy Blank Samples………………………….83

B. Observation Logs …………………………………………………………… 87

C. Student Work Sample: Grab and Count …………………………………….. 93

D. Student Work Sample: Number Writing ……………………………………. 97

E. Student Work Sample: AIMSweb Test of Early Numeracy……………….. 106

F. Sample of Research Based Activities ………………………………………118


Chapter 1: Introduction

Background

In the United States, children attending public school are instructed using a set of grade

level standards for English-language arts and mathematics. In 2009, Connecticut adopted the

Common Core State Standards (CCSS) as a way to ensure our students were college and career

ready upon completion of high school (Roberge-Wentzell, 2015). Along with the CCSS,

Connecticut began using Response to Intervention (RTI) as a way to target those students not

meeting the standards and demonstrated needing extra support. More specifically, mathematics

instruction shifted as the implementation of the CCSS and RTI began.

The shift in mathematics instruction has allowed for me to become a Mathematics

Instructional Coach and Intervention Specialist at the elementary level. This allows me to work

with educators on research-based instructional strategies for their entire class during Tier 1

instruction and work with students who’ve shown a need for Tier 2 or Tier 3 intervention.

However, as I enter my fourth year in this position, I’ve noticed that once students enter

intervention, it becomes very challenging to exit them back into the Tier 1 environment. Usually

this is due to students having too large of a gap in their early number sense to ever close the gap

between themselves and their peers. In order for RTI to truly be effective at closing the

achievement gap in mathematics, intervention must start earlier. This thesis study will take a

close look at the impact early numeracy instruction can have on the RTI process and individual

student growth.


Statement of the Problem

Creating a strong mathematical foundation for children in our schools today is not an

easy task. With various definitions of what number sense looks like and how it should be

addressed, it is understandable that educators would have a difficult time addressing number

sense needs in their classrooms. The literature review did determine that to create students who

would be successful in mathematics, they must have a strong understanding of number

(Faulkner, 2009; Fosnot, 2001; Griffin, 2004; Jordan, 2007; Sarama & Clements, 2009; Van de

Walle, 2013).

Knowing that students must have a strong number sense foundation before they can

master more challenging mathematics with addition, subtraction, multiplication, and division or

whole and rational numbers, there was very little research studies done that addressed RTI and

developing number sense. A key finding from a variety of intervention studies was that students

needed to have their understanding of number addressed at earlier ages (Browder et al., 2012;

Bryant et al., 2008; Douglass & Horstman, 2011; Fuchs et al., 2008; Yung & Reifel, 2010). What

the studies failed to document was whether or not developing interventions for kindergarten and

first grade students targeting their number sense abilities was more effective than providing

targeted intervention in later grades that did not address number sense.

Thesis Study

The purpose of this study was to explore the effectiveness of early numeracy intervention

with Kindergarten students. This study explored the following questions:

1. What is the impact of early numeracy intervention on Kindergarten students?


2. a) Does early numeracy Tier 2 intervention have a positive effect on the achievement

gap between intervention students as compared to their peers not receiving mathematics

intervention in Kindergarten?

2. b) Is there any difference in progress within the two groups?

This study aimed to focus its attention to the RTI model within mathematics and how to support

our youngest learners in their early numeracy development.

This single case study provided an opportunity to explore the impact of early numeracy

intervention on a small group of kindergarten students. For the purpose of this case study, the

researcher collected multiple measures of quantitative and qualitative data in order to further

analyze the impact of early numeracy intervention. When looking at the qualitative data collected

(i.e., observations, field notes, and documents) the constant comparative data analysis method

was used. Creswell (2012) described the constant comparative method as a way for the

researcher to identify categories and themes from the data. The researcher looked for areas of

positive growth in observations and student artifacts.

For the quantitative data (i.e., AIMSweb Test of Early Numeracy), an independent

sample t-test was used. The results from the case study were compared to a random sample of

their kindergarten peers whom did not receive early numeracy intervention to determine if there

was a significant difference between the groups. The researcher was able assess if the differences

between the two independent groups was more or less than anticipated for the general population

of students by using an independent t-test (Creswell, 2012). The researcher aimed to gain a

clearer understanding of the impact early numeracy intervention had on students showing a gap

in their number sense understanding at the start of kindergarten.


Summary

This single case study demonstrated that students who receive targeted intervention

surrounding early numeracy will make positive growth on early numeracy assessments and in

student work. Students received six weeks of Tier 2 early numeracy intervention along with Tier

1 mathematics instruction and were able to make strong gains as compared to their peers not

receiving intervention. This case study also revealed a closing in the achievement gap when the

intervention students were compared to their peers suggesting that early numeracy intervention

in kindergarten is the first step in remediating a gap in a child’s number sense foundation. The

researcher gained a clearer understanding of how positive early numeracy intervention can have

on kindergarten students when students received targeted instruction that matched their learning

needs.

Definition of Terms

Base 10 – Base 10 refers to the understanding a child has of a digit holding a certain value based

on its placement within a multi-digit number.

Cardinality – Cardinality refers to the understanding that the last number counted in a set

represents the entire quantity of that set.

Subitizing – Subitizing refers to a child’s ability to see a number within five without having to

physically count the set.

Quantity Discrimination – Quantity discrimination refers to an ability to determine the larger

number in a set of two numbers


Chapter 2: Literature Review

Introduction

Education has been constantly changing and adjusting to meet the growing needs of

learners throughout history. One major shift has been identifying the appropriate content and

strategies that should be addressed as children enter public education. In Connecticut, through

the adoption of the Common Core State Standards (2010) and Response to Intervention (RTI)

(2008) educators have provided educators with standards and processes to help meet the needs of

learners. This literature review will aim to define the core foundation of a child’s mathematical

understanding and how this is implemented through RTI. The literature was used to help answer

the question, what does effective mathematics intervention look like in Tier 1 and Tier 2?

To begin with, research was sought to explore what a child’s mathematical foundation

looks like in regards to number sense. Defining the term “number sense” was a difficult task as

was determining what researchers claim are the most effective practices at building a strong

number sense foundation. The literature then led to a discussion on what the current RTI

practices are in Connecticut in regards to Tier 1 and Tier 2 intervention. After defining what the

RTI process should look like, the purpose was then to find evidence of mathematics intervention

addressing a child’s number sense ability. For purposes of this study, Tier 3 intervention was not

explored as the goal was to determine if appropriate interventions were occurring within Tier 1

and Tier 2 to meet a child’s mathematical need so that they did not need to be referred to Tier 3

intervention services. The goals of this literature review were to define number sense, explore

effective mathematics instruction, and determine the impact of early intervention using what we

know about creating a strong number sense foundation on meeting a child’s educational needs.


To conduct this literature review, the methodology was to first to find qualitative and

quantitative studies were accessed in regards to how students learn mathematics. While

conducting a search review in academic databases the key terms that were searched were

mathematics instruction, best practice in mathematics, and elementary mathematics. Throughout

the literature that was read, the focus was then narrowed to the very beginning building blocks of

a child’s mathematical understanding which was defined as their number sense. In databases

such as Google Scholar, the key term number sense allowed for many research articles to be

used when defining a child’s number sense. Once the aspects of number sense were looked at,

studies were examined to understand if a child’ number sense impacted their future mathematical

understanding.

The viewpoint of this review then shifted to researching qualitative and quantitative

studies in academic data bases that incorporated mathematics intervention within Tier 1 and Tier

2 environments. Throughout the literature search process the objective was to find literature that

supported early mathematics intervention that addressed a child’s number sense in order to close

the achievement gap. The search began by identifying key terms such as RTI and identified a

few specific researchers that focused their intervention studies on mathematics. The literature

search demonstrated that there is not an abundance of mathematics intervention studies that

focuses on number sense nor research studies completed that tracked the progress of these

students. Upon reflection of the literature, mathematics intervention is an area that needs further

investigations into how to successfully build a learner’s sense of number and enable them to be

successful within mathematics through elementary school.


Number Sense

The first area to address begins with a child’s foundation for learning mathematics and

there is significant research in the area of number sense development to be examined (Faulkner,

2009; Fosnot, 2001; Griffin, 2004; Jordan, 2007; Van de Walle, 2013). To begin with, we must

think about what the term “number sense” means and how that relates to students’ abilities to

understand mathematics. For most teachers number sense has not clearly been defined, which

would make it a difficult concept for teachers to teach and assess (Faulkner, 2009). Number

sense requires the construction of a rich set of relationships between quantities in space and time,

counting numbers, and formal symbols and language used to describe an idea (Griffin, 2004).

The key word to look at within that description is “construction.” If students are only exposed to

mathematics as a relationship between numbers and rules, then they will never be able to

discover and explore relationships between quantities and numbers nor be able to express those

relationships in various ways. If educators cannot provide students with a solid number sense

foundation then it will be more difficult for students to generalize their understanding of whole

numbers to place value or their future work with integers and irrational numbers (Griffin, 2004).

Students who learn solely about rules lack the understanding of the meaning behind numbers.

Jordan (2007) identified number sense as the “intuitive knowledge of numbers” (p. 64).

This knowledge included the ability to discriminate and compare quantities, internalize counting

principles, and estimate quantities on a number line. Fosnot & Dolk (2001) stated that for

children to build their number sense they must understand that a number represents an amount.

When you count a group of objects, the last number you land on stands for the entire amount,

this is called cardinality. As you can see, there are various definitions of number sense that


makes it challenging for an educator to accurately address within a mathematics classroom.

Jordan (2007) claimed that without this solid foundation, students would have difficulties in

mathematics as they progress through school. Jordan also claimed that too often educators

believe a child’s mathematical disability is tied to their inability to memorize facts that are really

a deeper rooted issue of their undeveloped sense of numbers. Both Fuchs (2008) and Fosnot and

Dolk (2001) agreed with Jordan (2007), that number sense needed to begin being taught in the

earliest school years, such as preschool. This information will be critical when looking at

implementing mathematics intervention later in this literature review.

The knowledge a preschooler has of mathematics predicts their success into high school

(Clements & Sarama, 2011). If a child’s mathematical ability is affected so greatly by their early

childhood and primary grades of math education, why aren’t more educators focusing on

creating stronger number sense? Based on an early integration of number sense study by Jordan,

Glutting, and Ramineni (2009), it was determined that addressing number sense early had a

strong impact on a student’s later mathematical performance. Their study was conducted to see if

a classroom environment in kindergarten that was rich with number sense activities had a

difference in academic achievement in subsequent grades. This multi-year longitudinal study

found that number sense made a meaningful contribution to the success of student achievement

by third grade. It was determined that a child’s number sense was a powerful predictor of later

mathematics outcomes- both at the end of first grade and the end of third grade. The most

powerful aspect of this research study was the discussion on how most students who are having

difficulty in mathematics post third grade, can link their underlying misconceptions to a lack in

number sense development. If mathematics instruction is going to be effective in creating strong

mathematical thinkers, it must begin with building a strong number sense foundation.


Developing Number Sense

Many researchers and mathematicians see number sense through a slightly different lens.

For example, according to Van De Walle (2013) number sense was a good intuition about

numbers and their relationships. It developed gradually as a result of exploring numbers,

visualizing them in a variety of contexts, and relating them in ways that are not limited by

traditional algorithms. The critical components of number sense are numeration, base 10

proportional reasoning, equality, quantity magnitude, and form of a number (Faulkner, 2009).

Within these components there lie characteristics of what someone with a solid number sense

foundation looks like. These characteristics include: (a) fluency in estimating and judging

magnitude, (b) ability to recognize unreasonable results, (c) flexibility when mentally computing,

and (d) ability to move among different representations. It is, perhaps, these characteristics that

educators should look for in their students. If we are to say a child has difficulty in the area of

mathematics then we need to look back at their number sense foundation before we develop

strategic instruction to help move them in their individual learning journeys. All of these

characteristics describe a student who reasons within their mathematics, problem solve, and

make connections between mathematics concepts. Notice that in all of the research on number

sense, not one has labeled it as a child’s ability to compute numbers. Both Faulkner (2009) and

Van de Walle (2013) discussed that a majority of teachers wanting students to know their facts

and be able to follow algorithms to solve problems but that should not be the focus of our

instruction. If children are taught through hands on activities and have ample opportunities to

collaborate and share mathematical discourse then they will notice patterns that will inevitably

transcend into traditional algorithms later on in their numeracy experiences.


According to Burns (2007), a students’ number sense related to the ability to make sense

of numerical concepts and procedures. Students with good number sense can think and reason

flexibly with numbers, use numbers to solve problems, spot unreasonable answers, understand

how numbers can be taken apart and put together in different ways, and see connections among

numbers. The use of the word flexibility is often present when describing what number sense is.

This is an important word for teachers to see, use, and discuss. What does it mean when students

are flexible with their thinking? What does it look like in a first grade classroom when children

are flexibly deconstructing and constructing numbers? When examining what it might look like

for children to be demonstrating their number sense understanding we need to look at what the

National Council of Teachers of Mathematics (NCTM) consider this practice to be. NCTM

(2000) stated that number sense developed as students understood the size of numbers,

developed multiple ways of thinking about and representing numbers, used numbers as referents,

and developed accurate perceptions about the effects of operations on numbers.

Educators should be looking for students to demonstrate their thinking in multiple ways.

One key word that NCTM includes is accurate and the idea that we don’t just want children to

solve problems and reason with their thinking, but also we want children to develop accuracy

when working with number operations. Thinking about what number sense is, what it looks like,

and how it will affect student’s learning throughout elementary school is how this proposal’s

instructional models were developed. The Common Core State Standards (CCSS) in

Mathematics also support NCTM’s key components to how children learn mathematics

(McCallum, Zimba, & Daro, 2010).


Critical Components to Number Sense

Both Van De Walle (2013) and Burns (2007) have devoted their mathematical careers to

developing highly engaged lessons for classrooms where the focus is student centered, and not

teacher directed. Van de Walle’s (2013) research reflected many research-based

recommendations to help teachers develop high-quality learning activities for children. The

concept of subitizing is talked about frequently within Van de Walle and will be a focus within

this research study. It is important that children develop their abilities to “see” numbers.

According to Penner-Wilger et al. (2007), the ability to subitize and enumerate numbers is one

of the first skills in developing numerical abilities For example, when they roll and dice they are

able to acknowledge they rolled a five before actually counting the dots on the die. Children

learn how to subitize amounts through repeated exposure of engaged activities such as classroom

games. A teacher cannot tell a child to subitize. A teacher cannot offer a rote algorithm to help

students recognize number patterns. A child’s ability to be able to subitize quantities is a key

foundational piece in their number sense (Burns, 2007; Penner-Wilger et al., 2007; Van de

Walle, 2013). Without it, it makes comparing relationships within numbers as more than, less

than, and equal to more difficult. Ginsberg et al. (2005) agreed with Van de Walle’s (2013)

suggestions on the fact that activities to enhance a child’s number sense must include many

hands-on learning experiences. Using games with dice, opportunities to play and manipulate

numbers, and offering students opportunities to model mathematics will all help create a stronger

number sense (Ginsberg et al., 2005). Teachers believe that if a child cannot learn to recognize

the number pattern, for example, on a die then they are doomed and cannot be helped. Research

proves that through activities such as dot plate flash, learning pattern games, and fill the towers


that students can indeed grow their knowledge of being able to see numbers without actually

counting them (Van de Walle, 2013).

Clements (1999) discussed the reason behind why we need to teach and foster the

development of subitizing. He described two types of subitizing: (a) perceptual subitizing, and

(b) conceptual subitizing. Perceptual subitizing is recognizing a number without using other

mathematical processes. For example, a three year old can typically identify the number three as

it relates to their age. Usually by pre-school age students can attain this perceptual subitizing

from numbers one to three. It is the idea of conceptual subitizing that will be an instructional

focus of this research study. Conceptual subitizing plays an advanced-organizing role. This term

applies to people who “just know” the domino’s number pattern and can describe it as a

composite of parts and a whole. They can look at the domino pattern of the number 8 and

describe it as one eight and two groups of four. These people can view numbers and number

patterns as units of units (Steffe & Cobb, 1988). Other interesting aspects of subitizing that are

often not considered are finger patterns, rhythmic patterns, and spatial-auditory patterns.

Subitizing can be developed through looking at these various patterns. They may also help grasp

the attention and understanding of a non-traditional learner. The research behind subitizing has

really shown that it is a key component and without this ability children may become

conceptually handicapped in learning arithmetic processes (Clements, 1999; Fosnot & Dolk,

2001; Steffe & Cobb, 1988). Children often use counting and patterning abilities to develop

conceptual subitizing. The more advanced a child’s ability is to group and quantify sets quickly

supports their development of number sense and arithmetic abilities (Clements, 1999).


Base 10

Another key element of research has identified that children with mathematics difficulties

begin using their fingers later and depend on them longer (Jordan, 2007). Teachers in grades

three and up are constantly noticing children whom are still counting on their fingers for basic

math. One reason behind this is a gap in their ability to see numbers (i.e., subitizing) and also

their ability to visualize the quantities and apply learned patterns. Various mathematics educators

believe that number sense refers to the intuitive knowledge of numbers. That is innate and cannot

be taught past a certain age. Children are either able to identify the last number they counted in

the set to be the total amount they counted or they can’t (Berch, 2005). If educators can identify

the break down in number sense early enough (i.e., prior to third grade) then it is absolutely

possible to “patch” and build strong mathematicians.

Teachers will frequently have students in upper elementary grades and say, “They just

don’t get how to subtract. They can’t carry or borrow. They just don’t know their facts. They just

can’t do fractions.” These ideas of what students can and cannot do in math tend to be about

students who have gaps in their number sense foundation (Jordan, 2007). If they had been given

the explicit instruction needed at an earlier age then they would be more successful with more

complex problems that arise in later grades. These students may have difficulty following the

“carry the one” notion in addition because they cannot visualize that the one really represents a

ten but all they are seeing is a one. What that student now needs is to go back to the concept of

place value and construct their own meaning of how ones group together to form tens. That

learner needs to be taught at a concrete level first before being continuously asked to show their

understanding at an abstract level. To do this, educators need to encourage cognitive strategies


such as cueing prior knowledge and following a Concrete-Representational-Abstract (CRA)

approach to learning (Geller & Smith, 2004).

Teachers do not intentionally try to teach children a topic they know is too difficult,

however, many educators do not know where to go or what to do if a child’s understanding at the

abstract level is weak. Educators need to utilize early numeracy assessments that focus on what

Jordan (2006) identified as the common core areas of misconceptions: counting sequence,

quantity discrimination, number transformation, estimation, and extending number patterns.

According to Jordan, these were skills that preceded a child’s ability to conceptually understand

mathematics in later years. One way to develop this number sense in young children is by

allowing them to explore and construct their own understandings within meaningful contexts.

Authentic contexts for word problems, games, and planned investigations can all be used as

effective early numeracy instruction inside a mathematics classroom (Fosnot, 2001). This

misconception that some children just “can’t do math” really stems from the fact that students

have never been taught in appropriate modes to be able to understand the math. It is just not

enough to be able to “do” the math.

This leads to research and discussion about place value and the background on what our

base 10 system is and how to convey that to our students. In kindergarten and first grade students

are exposed to counting numbers up to 100 and the Connecticut CCSS have first grade students

counting to 120. They begin as early as five years old experiencing possibly groupings of such

numbers and learning how to count by tens. Ross (1989) identified five levels of place value

understanding. These levels naturally build upon each other beginning with children identifying

individual numerals to follow understanding that multi-digit numbers are composed of different

groupings of ones, tens, etc.. Battista (2012) presented a trajectory of a children’s early number


foundation that began with counting and built up through understanding place value. His view

was that place value was more than just identifying the value of numbers but in utilizing

counting skills like skip-counting by tens, hundreds, and thousands that helped initially build a

child’s place value understanding. Ross (1989) and Battista (2012) differed in their views of how

place value was developed, but both identified counting as a critical component for a child to be

successful.

Usually in second grade there is a heavy emphasis on place value and a students’ ability

to group ones into tens and tens into hundreds and hundreds into thousands. All too often a

teacher “shows” or tells students how to bundle or group their ones and tens. There is little

construction of the concepts by students. A number of research studies show that no matter how

many times children were told for the number 62 that “six is in the tens place and six is in the

ones place” students still misidentify 62 for 26 (Cooper & Tomayko, 2011). Cooper and

Tomayko described their colleagues, Kamii and Joseph’s (2011), findings that in fact when

presented with the number 16, most first graders and second graders could not identify the one as

a ten. They viewed it as simply one unit. Their research suggested that this misconception is

prevalent up through fourth grade in many cases of students whom are struggling in

mathematics.

This dilemma brings us back to teaching number sense. How can a nine year old still not

see that the number 16 is composed of one ten and six ones? This is because their understanding

of unitizing was not solid. Unitizing is the preliminary place value understanding that ten can be

represented and thought of as one group of 10 or 10 individual units (Fosnot, 2010). Once you

are confident that students can count to and past 100, you can begin working with them on

groupings of ten to begin building this idea of unitizing. Often in grades two or three you might


see students solve 35 +67 = 9 12. This is because they are not seeing the difference in the value

of each place and are not able to group past individual units of one. Van de Walle (2013)

discussed that we don’t want to force children in seeing the number 53 as just five tens and three

ones but that we can quantify the number into groups of five tens and three ones. We need to

make sure children understand we are not changing the number but counting it in a different

way. Students who struggle with this concept might look at the grouping of 53 in five tens and

three ones and say it’s 53 but when you “undo” the grouping they will not see it as the same

number and have to count the 53 by ones. This type of instruction where children look at ones

and compare them to groupings of tens as the same quantity of numbers will be an instructional

focus of this research study to help build their sense of unitizing.

Van de Walle (2013) created many hands on activities where students construct and

deconstruct groups of tens that build on one another from counting, to place value notation, to

base-ten models, to finally developing base 10 concepts. Battista (2012) discussed that when

students finally develop base 10 concepts that they replace counting strategies with “combining

and separating numbers by place value parts.” (p. 3). Van de Walle (2013) suggested that

teachers often jump to teaching children the base-ten concept before all of the background work

has been in place. This is one reason why many children get to fourth grade and still have gaps in

their understanding of place value.

A major piece of number sense is the idea of flexibility. We see this again when

examining place value. Research done by Faulker (2012) showed that telling students in example

the number 384, that the “4” is in the ones place, the “8” is in the tens place, and the “3” is in the

hundreds place is not an effective strategy to helping them learn place value. Instead, we need

students to be flexible and group the number 384 in many ways leading to the exploration of


different values of numbers. Van de Walle (2007) described allowing students to break apart

numbers in many ways. For example, 256 can be one hundred, 14 tens, and 16 ones but also 250

and six. Taking numbers apart and recombining them in flexible ways is a significant skill for

future computation.

Hands-on Approach

We can teach number sense. In fact, we can provide students many opportunities to

develop and foster their mathematics knowledge. Educators must provide students rich activities

for making connections, exploring and discussing concepts, and ensuring an appropriate

sequence of concepts (Griffin, 2004). This action research in itself will help prove that with

hands-on activities and allowing students time to construct their own understandings that their

number sense will improve. Many teachers believe that if a student can perform well on an

assessment then they have done the appropriate learning.

With the adoption of the CCSS it is believed that new assessments will contain perhaps

the most important piece of mathematics that has not traditionally been tested. That is, the how

and the why behind a students’ thinking. The only way to have students internalize the how and

why is to begin by building their number sense with the use of concrete manipulatives.

Manipulatives are concrete tools used to create an external representation of a mathematical idea

(Burns, 2000). If you teach students to simply stack the numbers on top of each other, line them

up, and then add, what have you really taught them? Asking children to first use math tools in

ways that make sense to them (i.e., play in pre-K and Kindergarten) you will have students later

on that can explain to you that they added by combining two (or more) groups together and

looking at the new whole group total. We might look at that statement as very basic but it is a far


better understanding of what addition is and how they came to their result instead of telling you

they just lined up the numbers.

In relating the students’ understanding of the base ten system, we have to look at the use

of manipulatives. One study by Cady and Hopkins (2007) showed the use of connecting cubes to

be crucial when teaching place value. Students could count these multilink cubes by ones,

connect them in tens, and then break them apart again to help develop patterns of building ten

ones and one ten. Before using the pre-made base 10 blocks it is a good idea to use cubes of

objects that can be put together in tens and taken apart. This will reinforce the idea that students

aren’t constructing a totally new number yet just building the same number is multiple ways

using groupings of ten. One study done on the impact of using base 10 blocks for first and

second graders suggested that students do not need to construct the understanding of 10 ones

being one ten before using base 10 blocks (Fuson & Briars, 1990). In this study, students used

primarily base 10 blocks and written number-word notation to add and subtract multi-digit

numbers. The study showed that as long as children know which base 10 block represents which

value that students can go about trading (i.e., carrying and borrowing) and are able to verbalize

their process. This study found a significant difference with students’ pre- and post-test scores

after having only instruction of place value using base-ten block manipulatives. However, later

research in later decades does suggest that the use of many different manipulatives can help

children be even more successful in their ability to transcend what they know about place value

into efficient addition, subtraction, multiplication, and division strategies (Cady & Hopkins,

2007). Cady and Hopkins had determined that using multi-link cubes were vital in a student’s

construction of the base 10 system prior to using base 10 blocks. This study concluded that it was


in the construction of the base 10 block that students gained a stronger understanding of five

which was a precursor to fully developing an understanding of ten.

A study by Swan and Marshall (2010) revisited the idea of what a mathematics

manipulative is especially when there are so many virtual tools readily available to most

classrooms and learners. A mathematic manipulative used to be defined as “concrete models that

incorporate mathematical concepts, appeal to several sense and can be touched and moved

around by students” (Hynes, 1986, p. 11). Marshall and Swan (2010) believed, however, that

mathematics manipulative cannot simply be used during hands-on activities but that students

should interact and engage with the math tool. They define mathematics manipulative as an

object that can be handled by an individual in a sensory manner during which conscious and

unconscious mathematical thinking will be fostered. The research done by Marshall and Swan

better suits this research study. Students can use manipulatives that may lead to development of

concepts and are used in purposeful ways. Neither definition, however, included virtual

manipulatives. In the age of mobile applications students should be exposed to mathematics in

this nature that is natural for them (McQuillan & CSDE, 2008). As technology such as the iPad

become more available for students, the use of virtual manipulatives will increase. With

technology, the base ten block could be broken down into single ones and re-built as the physical

blocks cannot.

In 2013, a study was conducted on students with Autism Spectrum Disorder (ASD) to

determine whether there were more benefits to physical manipulatives than with virtual

manipulatives (Bouck, Courtney, Doughty, & Satsangi, 2013). The study had a very small

sample size of only three students, but was able to identify that all three students “demonstrated

greater accuracy and faster independence with virtual manipulatives as compared to the concrete


manipulatives” (p. 190). Even though this was done with students with ASD, it should be

questioned whether this personal action research will demonstrate the same findings. Similar to

Bouck et al. (2013), Puchner, Taylor, O’Donnell, and Fick (2008) also found that using

manipulatives help learners to internalize their mathematical representations. Their study

included 33 K-8 teachers embarking on a summer intensive professional development program

that taught how to successfully incorporate manipulatives into their mathematics instruction.

Their conclusions demonstrated that teachers whom incorporated hands-on learning through

manipulatives in carefully designed lessons observed an increase in their understanding of

number and operations. As discussed early in this review, students need opportunities to

construct their own understandings of number and using manipulatives is one way to accomplish

that goal.

Response to Intervention (RTI)

What happens when a child is showing deficits in mathematics? In 2008, the Connecticut

State Department of Education (CSDE) presented a document titled, Using Scientific Research‐

Based Interventions (SRBI): Improving Education for All Students: Connecticut’s Framework for

Response to Intervention (McQuillan & CSDE, 2008). The RTI process in Connecticut clearly

defined what tiered instruction should look like. Tier 2 interventions are short term (e.g., eight –

20 weeks) and remain part of the general education system with supports from specialists.

Interventions must be research‐based as much as possible, be reasonably feasible for educators to

use, and accurately target the student’s area(s) of difficulty. These interventions are supplemental

to the core academic instruction that is delivered in the classroom by the classroom teacher or

other specialists. These interventions do not replace core instruction, nor do they remove

responsibility for the child’s learning from the classroom teacher; rather, students receive support


both in Tier 1 and Tier 2. If appropriately matched to individual student’s needs and

implemented with fidelity, interventions should result in growth for most students receiving Tier

2 interventions.

According to McQuillan and the CSDE (2008), the basic components of the RTI model

are:

1. The assumption that scientific research should be used to inform educational

practice as much as possible.

2. A belief in collective responsibility, accountability and the power of education.

3. A willingness to be transparent with a relentless focus on continuous

improvement.

4. A focus on prevention and early intervention.

5. School wide or district wide high quality core curriculums, instruction and

comprehensive social/behavioral supports.

6. Monitoring fidelity of implementation.

7. A comprehensive assessment plan with universal common assessments and

progress monitoring.

8. Data driven decision making with clear decision rules. (pp. 16-20)

The CSDE has outlined what the goals and processes of RTI should be in a school setting. What

the CSDE has not shown is how that relates to mathematics intervention and if implemented

properly, what the benefits will be for students struggling with mathematics.

Response to Intervention in Mathematics

While the big idea of number sense has not been clearly defined, there were critical

components that are part of a child’s early number sense foundation. We do know that there are


research based instructional strategies that can help improve a child’s early numeracy foundation

and those will be explored and used in this action-research. In RTI, it is imperative that data be

used to identify a student’s weakness and design a support system that will improve that child’s

mathematical ability (Powell & Stecker, 2014). Data-based individualization is a continuous

process that links assessment, instruction, and intervention (Fuchs et al., 2008). It is the RTI

model that will allow this research to take place. In an era focused on standards based testing, it

has been important for students not meeting benchmark standards to receive extra support.

The notion of RTI can be described as educators looking at a student’s responsiveness to

high-quality instruction before looking for an organic disability (Koellner et al., 2011). It is

important to identify a child’s specific need on the mathematics learning trajectory in order to

develop targeted instruction through the use of formative assessments (NCTM, 2000). The RTI

model allows for students to be pre-assessed and look at their current math understandings so

that instruction for each learner can be differentiated and build off of what they already know.

Good formative assessments can help teachers and RTI specialists reveal what students

understand, what invented strategies they use, and where their misconceptions are (Koellner et

al., 2011). A response to intervention teacher is responsible will for delivering research-based

high quality instruction and closing gaps in student’s mathematical understanding. RTI is to be

used as purposeful instruction and not focused on simply skills and procedure to perform better

on standardized testing.

Many RTI approaches systematically provide instruction to students in a pull-out method,

but one study done with schools in Ohio indicates that effective RTI instruction can happen in an

inquiry-based classroom (Douglas & Horstman, 2011). This study also identified students in

three tiers as does Connecticut’s SRBI system. Research-based strategies can be used to help


students fill gaps and create a deeper understanding, and can give students the tools to be more

successful. Douglas and Horstman identified that most RTI specialists are serving a wider range

of needs than a regular education teacher but that they don’t need to service them outside the

classroom. It was discovered that before students were identified as kids who needed to go into

Tier 2 or Tier 3 pull out instruction that they were already significantly behind. It was a “wait to

fail” model where regular education teachers were not necessarily differentiating their instruction

in Tier 1 because once students “failed” for long enough they were pulled out and no longer an

issue for the regular education teacher. This “wait to fail” model,” in my opinion, is a flaw in the

RTI process. Without early identification and intervention before a student is falling significantly

below grade level standards, many students will continue to have mathematical deficits through

out their school years (Bryant et al., 2008).

Douglas and Horstman (2011) also described the tiered system of interventions as simply

adjustments of intensity of instruction in order to meet individual needs. The authors concluded

that in inquiry-based classrooms teachers can easily and more readily differentiate instruction

therefore meeting the needs of learners in a more efficient way and not waiting for students to

fail to receive such specialized instruction as in the RTI model. For this to exist, mathematics

intervention needs to begin in the early primary grades. Research has shown that students

develop their core number sense starting in infancy through the primary elementary school

grades (Sarama & Clements, 2009). While there is significant research in strengthening a child’s

number sense (Faulkner, 2009; Fosnot, 2001; Griffin, 2004; Jordan, 2007; Sarama & Clements,

2009; Van de Walle, 2013), the question becomes how does this impact mathematics RTI?


Evidenced-Based Decision Making

Providing intervention services to students not meeting benchmark standards must

receive research-based best practice. One method that was studied by Fuchs and Kearn (2013)

was Cognitively Focused Instruction (CFI). Fuchs and Kearns explored how skills based

instruction leads to failure among students in need of mathematics intervention. If a student has

not responded well to skills-based Tier 1 instruction in their classrooms then it is common sense

to use a different approach when the student enters Tier 2 intervention. CFI offers an approach to

instruction that targets the child’s learning deficit such as their working memory in order to build

on their number sense foundation. CFI is also well suited for the most intensive Tier 3

interventions as it can be individualized. In order for CFI to be successful, educators must adjust

practices accordingly to each student. This means using data that determines a child’s level of

performance and then determining which relevant strategies and techniques will assure that

children increase their levels of mathematical understanding (Berch & Mazzocco 2007; NCTM

2000). In order to provide the best-suited type of intervention, the educators involved must target

the exact areas of instruction.

In an analysis done by Fuchs and Kearns (2013), they researched CFI that targeted

cognitive processes such as working memory, metacognition, language/auditory processing,

processing speed, and visual processing. They determined that when students received individual

CFI versus skills based interventions, they were able to improve their academic success and

make progress on closing the achievement gap with their peers. The authors were not clear on

how individual CFI was identified which shows that there is future research needed on this and

how it can be paired with research-based instruction strategies that improve a child’s number


sense. The NCTM (2000) suggested that in order to make the appropriate decisions regarding a

student’s mathematical ability, formative assessment measures must be used to guide the RTI

process and assist in designing an effective RTI plan (Koellner, Colsman, & Risley, 2009).

It appeared that extensive data must be collected on a learner before the RTI process can

begin. Formative assessments can reveal what students know and what they are struggling with

(Koellner, Colsman, & Risley, 2009). Along with formative assessments, the intervention

process also needs to use more individualized diagnostic assessments that help narrow the target

areas of instruction. Fuchs (2008) emphasized the use of diagnostic assessments that can also be

considered Curriculum-Based Assessments (CBA) and can be in the form of a student interview,

individual concept work sample, skills assessment, or any piece of student work that directly

connects to the mathematics curriculum. Educators use various CBA’s to determine a child’s

strength and for error analysis to determine areas of weakness (Gresham, 2009). The research

showed that multiple measures of assessment were needed in order to design and implement the

most targeted intervention plan. Educators must use standardized assessments, formative

assessments, and diagnostic assessments as a means to provide evidence-based interventions

(Koellner, Colsman, & Risley, 2009).

Implementing Mathematics RTI

Once educators have diagnosed a student as needing mathematics intervention, how will

their plan be implemented? One notion is that the intervention can occur in an inquiry-based

mathematics classroom. Having an interventionist work with one student on a one-to-one basis is

often not practical in our current public schools (Douglass & Horstman, 2011). A key component

of effective intervention is diagnosing a student early and providing tiered intervention that

begins with the Tier 1 classroom environment (McQuillan & CSDE, 2008). In the Tier 1


environment, teachers can use small groups to address various needs among students. In these

small groups it would be possible to address various needs from students who struggle with

counting quantities accurately to being able to solve multi-digit word problems. In inquiry-based

classrooms, the general teacher must be able to hear and interpret what students are saying, be

able to skillfully probe when the student is not clear on an understanding (Douglass & Horstman,

2011).

Tier 1. In a Tier 1 mathematics environment, the teacher needs to provide students with

plenty of rich and authentic opportunities to encourage a range of student thinking and

questioning (Lampert, 1990). When students have opportunities to engage in mathematical

discussions, they are able to share what they know and how they know it. Communication has

been recognized as a critical component of developing a child’s mathematical understanding,

thinking, and problem solving skills (Yung & Reifel, 2010). In the study of kindergarten students

conducted by Yung and Reifel, they determined that in an inquiry-based classroom it was

possible that a teacher could integrate hands-on learning and plenty of opportunities for children

to share their mathematical understandings, even at such a young age. This was an example of a

Tier 1 classroom approach that enabled the teacher to meet individual student needs based on the

communication or “math talk” that was observed during inquiry-based lessons. It is in this type

of environment that can make Tier 1 mathematics instruction highly effective.

Tier 2. According to the McQuillan and the CSDE (2008), when a child is not making

adequate progress in the Tier 1 classroom, a second tiered approach to instruction must be

implemented (i.e., Tier 2). This is defined as short term additional interventions (i.e., eight-20

weeks) and do not replace the core curriculum (McQuillan & CSDE, 2008). If these

interventions are appropriately paired to a student’s instructional need then the child should


begin making progress to grade level standards. Tier 2 intervention in mathematics is put in place

to prevent a student’s further math difficulties with the use of ongoing progress monitoring and

consists of small group, explicit mathematics instruction (Bryant et al., 2008). According to

Bryant and her colleagues this type of intervention must incorporate the use of CRA sequencing

and be evidence based. This type of instruction engages students in the process of exploring

mathematical ideas and then leads them to represent their thinking and finally be able to

internalize the concepts.

The goal of a Tier 2 intervention plan is to help the learners begin achieving at the same

standards as their peers. However, the trend appeared to be that by the time students were

receiving Tier 2 mathematics intervention, their gaps in understanding were too large to be filled

in such a short time frame when students present with number sense deficits in later elementary

grades (Bryant et. al., 2008). It becomes quite impossible to improve a third grader’s ability to

divide if the student is still struggling to understand, for example, how to compare basic numbers

based on their place value representations. Mathematics intervention needs to begin in the early

grades in order to improve on their number sense abilities before it is too late and the academic

gap widens (Clements & Sarama, 2004).

Bryant et al. (2008) conducted a study looking at the effects of early intervention with

grade one and grade two students. The students participated in instruction that included number

sense concepts such as number concepts, place value, and early computation. Using a range of

progress monitoring and assessment tools, it was concluded that a case should be made for Tier 2

mathematics intervention to include number sense concepts because it is such a large predictor

for future math difficulties. At the end of the study, there was a smaller gap between the grade

one students receiving intervention as compared to their peers than was observed with grade two


students. This only continues to support the idea that students whom are in danger of falling

behind in mathematics need mathematics intervention earlier rather than later (Fosnot, 2001).

Browder et al. (2012) found that students in grades three through five may need early

numeracy instruction when having math difficulties but that they also need age appropriate

content as well. “For example, a 10-year old student needs not only to comprehend the concept

of the numeral 4 but also be able to apply it to the activities of a typical fifth grade class.” (p.

309). How does an interventionist or certified teacher address the needs of early numeracy gaps

while also attending to grade level content of children in the intermediate grades? Browder

suggested using sequential lessons within a curriculum design that begins with numeracy

foundations first and progresses into more challenging topics. Whereas Fuchs (2008) suggested

that students needed to use what they were learning from their tiered interventions on a daily

basis. This could prove to be challenging if a learner is working on unitizing during intervention

but is learning how to decompose fractions in their regular Tier 1 environment.

Educators and interventionists use rich amounts of data to provide tiered interventions

that should enable students to be more successful in mathematics but the dilemma becomes

whether or not to build on from where a child first presents a weakness or help support them with

current grade level skills? Lembke, Hampton, and Beyers (2012) suggested that in order for the

systematic instruction of Tier 2 intervention to be successful, educators must build a students’

mathematical understanding in a logical order (i.e., place value before addition). Along with this

Gersten et al. (2009) explained that the instruction within a Tier 2 intervention should consist of

step-by-step modeling of how to solve basic math problems, computation, and be paired with

problem solving and reasoning behind their math. When the best intervention model takes place


where students are being systematically taught at their identified instructional needs, is Tier 2

intervention effective?

Summary

Creating a strong mathematical foundation for children in our schools today is not an

easy task. With various definitions of what number sense looks like and how it should be

addressed, it is understandable that educators would have a difficult time addressing number

sense needs in their classrooms. However, this literature review did determine that to create

students who would be successful in mathematics, they must have a strong understanding of

number (Faulkner, 2009; Fosnot, 2001; Griffin, 2004; Jordan, 2007; Sarama & Clements, 2009;

Van de Walle, 2013).

Knowing that students must have a strong number sense foundation before they can

master more challenging mathematics with addition, subtraction, multiplication, and division or

whole and rational numbers, there was very little research studies done that addressed RTI and

developing number sense. A key finding from a variety of intervention studies was that students

needed to have their understanding of number addressed at earlier ages (Browder et al., 2012;

Bryant et al., 2008; Douglass & Horstman, 2011; Fuchs, 2008; Yung & Reifel, 2010). What the

studies failed to document was whether or not developing interventions for kindergarten and first

grade students targeting their number sense abilities was more effective than providing targeted

intervention in later grades that did not address number sense. Many of the studies discussed in

this review identify positive conclusions when a student’s intervention was targeted to their

specific need of number sense or place value, but never discuss if in the long term, the

intervention was effective.


Thus, leads to the basis of my purposed action research. If educators address a gap in a

child’s mathematical understanding at an earlier age, will that student be able to close the gap

with their peers and return to Tier 1 instruction? What I currently observe in my job as a

mathematics educator is that when students begin receiving intervention too late, the gap

between their conceptual understandings of grade level content as compared to their peers is just

too large to adequately address in a typical Tier 2 intervention. Further research is needed to

begin exploring the differences and benefits from early intervention that targets a child’s number

sense ability as compared to intervention that occurs later in the intermediate grades.


Chapter 3: Research Purpose and Questions

Research Questions

The research that was reviewed, indicated that early numeracy was a deficit in many

students requiring mathematics intervention as they progressed through elementary school. As

such, the purpose of this study was to explore the effectiveness of early numeracy intervention.

This study explored the following questions:



gap between intervention students as compared to their peers not receiving

mathematics intervention in Kindergarten?


The study focused on one single group of Kindergarten students receiving early numeracy

intervention. The researcher aimed to make meaning from multiple modes of data collection

about how early numeracy intervention impacted their mathematics ability. There was also

comparisons made if there were any findings within the small intervention groups in relationship

to their progress as compared to their peers not receiving intervention. For these reasons, the

methodology used for this study was a single case study.

Methodology

A case study will allow the researcher to collect multiple modes of data to track the

progress of Kindergarten students receiving Tier 2 mathematics intervention. Creswell (2012)

indicated that a case study will allow a researcher to gain an in-depth understanding of the case

through collecting numerous types of data. This methodology will also allow the researcher to


examine the impact a set of early numeracy intervention activities has on a small group of

kindergarten students during the course of an eight-week intervention cycle. The critical

components of number sense are numeration, base 10, proportional reasoning, equality, quantity

magnitude, and form of a number (Faulkner, 2009). Within these components there lie

characteristics of what someone with a solid number sense foundation looks like. These

characteristics include: (a) fluency in estimating and judging magnitude, (b) ability to recognize

unreasonable results, (c) flexibility when mentally computing, and (d) ability to move among

different representations (Van de Walle et al., 2013). These are the components of early

numeracy that the researcher will use the case study design to explore.

The researcher will use the case study methodology because the intent will be to learn

about the impact early numeracy intervention may have on Kindergarten students and how that

group of students compared to those of their peers not receiving intervention. A case study also

lends itself to work within a constructivist theoretical framework. A constructivist framework

will allow the researcher to use her perspective when analyzing the multiple pieces of data from

the small group and be able to construct the impact early numeracy intervention had on those

students. According to Creswell (2003), a constructivist approach will allow the researcher to

make meaning from their selected experiences. These derived understandings are completely

subjective to the researcher and the work they are analyzing. Being able to look deeper at the

early numeracy development of Kindergarten students and seek to understand the impact of the

intervention will be a subjective process and relies on the experiences of the participants. Crotty

(1998) stated that “it is clear different people may construct meaning in different ways, even in

relation to the same phenomenon” (p. 9). The conclusions and findings of this case study will be


constructed by the researcher and others may possibly interpret the data in different ways, thus

proving there is not one exact answer to the questions in this study.

Finally, Yin (2003) indicated that a case study approach should be taken when you want

research contextual conditions because you believe they are relevant to the phenomenon under

study. In this case, this study will look to understand how early numeracy intervention impacts

the achievement gap within the context of Kindergarten students. Yin also defined three different

types of case studies; explanatory, exploratory, and descriptive. This study will be defined as an

exploratory case study because it will explore early numeracy intervention within a small group

and it does not have a single set of outcomes. This will be a single case study because it will look

at one group of students in one setting (i.e., local elementary school) and the intervention will be

delivered by the researcher. It is the intention of this researcher to be able to make

generalizations about the impact early numeracy has on Kindergarten students through a case

study design model.

The Case

This single case study was conducted in the suburban town of South Windsor,

Connecticut within the elementary setting of Pleasant Valley Elementary School. The

participants were a small group of kindergarten students who will ranged in ages from four to

five years old. This school contains roughly 350 students. In 2013, 92% of the kindergarten

students attended pre-school which meant most of our kindergarten students started the school

year with a foundation in early literacy and numeracy skills. The small group of students chosen

to participate in this case study were the five students that scored the lowest on the fall

AIMSweb Test of Early Numeracy (TEN) Assessment.


This case study began during September 2015 with all incoming kindergarten students

being screened using the AIMSweb TEN. This assessment helped the researcher identify the

bottom six students who performed below grade level standards on the four subsections of the

assessment: oral counting, quantity discrimination, number identification, and missing number

sequencing. Those students then spent four weeks receiving Tier 1 mathematics instruction and

were re-tested using the AIMSweb TEN to determine the amount of growth that occurred while

not receiving early numeracy intervention. Those five students then received early numeracy

intervention in a small group setting four days a week for 30 minute sessions. The intervention

cycle lasted approximately six to eight weeks depending on the amount of progress being made.

At the end of the intervention cycle, the AIMSWeb TEN was administered one last time to use as

quantitative data for determining growth and progress achieved. A random sample of five

kindergarten students also took the AIMSweb TEN at this time whom have not been receiving

early numeracy intervention. This served as comparative data to further explore the impact early

numeracy intervention has on students.

Data Collection Methods

For the purpose of this case study, the researcher will collect multiple measures of

quantitative and qualitative data in order to further analyze the impact of early numeracy

intervention. The data collection methods that will be utilized are outlined in Table 3.1.

Table 3.1

Data collection methods by research question

Research Question Method(s)

1.What is the impact of early numeracy ● Artifacts, student work samples,


intervention on Kindergarten students? intervention documents

● Observations

● AIMSWeb TEN

2 a) Does early numeracy Tier 2 intervention

have a positive effect on the achievement gap

between intervention students as compared to

their peers not receiving mathematics


2. b) Is there any difference in progress within

the two groups?

● AIMSWeb TEN

● Artifacts and student work samples

Each of these methods of data collection is described individually in the following sub-sections.

Artifacts and Documents

This research study looked at the type of early numeracy instruction kindergarten

students are involved in during a Tier 2 intervention. Creswell (2012) stated that the researcher

should be able to identify the types of documents that will aid in answering the qualitative

research questions. To address both research questions, the researcher collected various artifacts

of student work, visuals, and intervention logs which document the early numeracy activities

students are participating in. These artifacts provided evidence as to the learning the students

involved in the case study are doing. It is the goal to be able to track their progress on their

student work over the course of time the intervention is being applied. The artifacts collected will


allow for a more comprehensive analysis to be done in terms of the impact of early numeracy

intervention.

Further, Sandelowski (2000) suggested that whatever is observed while in the field,

should be documented and described in order to identify common key findings. Collected student

work samples, photographs of hands-on learning, and intervention documents will help the

researcher describe the impact the intervention is having on the students participating in the case

study. Devers and Frankel (2000) stated, “when the study is more exploratory or attempting to

discover and/or explore theories and concepts, a very open-ended protocol is appropriate to

consider” (p. 336). The purpose of this research study is to explore the impact of early numeracy

intervention, which according to Devers and Frankel, would suggest that documents and artifacts

can be collected as needed throughout the study. There is no specific protocol as to exactly

which documents will prove useful when interpreting the impact of early numeracy instruction,

however, it will still be imperative that student work samples be collected and analyzed to form a

greater understanding of each learner within the case study.

The types of documents that will prove useful will be each participant's mathematics

journal that they will use to document their learning in. For Kindergarten students, the type of

work the researcher is likely going to see would be number writing, pictures to represent

computation strategies, and decomposing numbers into friendly parts. The other piece of

documentation that will be collected will be visual images (e.g., photographs) of student work

and student participation in hands-on learning such as subitizing activities. The advantage of

using visual documents within this case study is that the images will provide real-life

documentation of a student’s learning versus simply relying on observational notes and student

work that may not make sense to the general population (Creswell, 2012).


Observations

The researcher will also use observations as a mean to collect data. These observations

made on each student and the learning behaviors they are exhibiting will aid in the lesson

planning and reflective practice needed to provide highly effective early numeracy intervention.

The observations will be firsthand information about the learning behaviors of the Kindergarten

students. As Creswell (2012) noted, collecting observations enables opportunities to document

information as it occurs, and to be able to study individuals verbalizing their thinking. These

observations will be made on weekly intervention logs and will have anecdotal notes on each of

the participating students. The intervention log is self created and allows room for learning

targets, dates of intervention, and observations for each child. Over time, the researcher will be

able to use the observations to begin making generalizations on the impact early numeracy

intervention has had on these students.

The role of the researcher will be as a participant observer. Creswell (2012) described a

participant observer as someone who takes part in the activities that they are then making

observations on. The researcher will be delivering the early numeracy intervention to the small

group of Kindergarten students, work with hands-on activities with the students, and then make

field notes at the end of each intervention session. Adler and Adler (1994) discussed that many

difficulties in ensuring observational data is both valid and reliable. Adler and Adler stated that

“naturalistic observation yields insights that are more likely to be accurate for the group under

study” (p. 5). Since the observations will be made directly from the researcher and implementer

of the instruction, the observations should provide useful data when looking to explore what

impact early numeracy intervention is having on students.


The observations will also take place in the same setting daily and recorded by the same

person. Creswell (2012) also suggested conducting multiple observations on each individual to

better understand each participant. Observations will also begin broad and begin to narrow as the

intervention continues and the researcher gains more insight on the participants. As

recommended by Altheide and Schneider (2012), observations will be made directly after each

intervention session takes place. Althneide and Schneider also suggested that notes be written

down not in the presence of the participant and that quick one word notes could be written down

by the observer while the intervention is taking place. After the intervention, the researcher will

use the intervention log to jot down important observations made on each participant. The

observations made will focus on the critical components of number sense, such as subitizing, the

construction of a rich set of relationships between quantities, counting numbers, and being able

to express numbers in various ways (Griffin, 2004). Since the observer will be an active

participant in delivering the early numeracy intervention, it will enable the observer to document

what the students have done, verbalized, and learned in each session (Althneide & Schneider,

2012).

AIMSweb Test of Early Numeracy

The quantitative method of data collection will be through a commonly accepted measure

within the school district this case study will take place in. The assessment is called the

AIMSweb TEN. There are four components to this assessment and each one is given in one-

minute timed intervals. The four assessments are oral counting, number identification, quantity

discrimination, and identifying missing numbers within a counting sequence (Clark & Shinn,

2002). Each task is designed to target one critical area of a child’s developing number sense

which is directly linked to both research questions within this study.


The AIMSweb TEN will be administered at the start of the case study to all Kindergarten

students. This method was chosen because it will allow the researcher to measure student

achievement, assess individual students’ abilities, and begin forming a picture of their strengths

and weaknesses (Creswell, 2012). The assessment will be used to determine those six students

who scored in the lowest percentiles for each task. After four weeks of Tier 1 classroom

mathematics instruction, the AIMSweb TEN will be administered again prior to beginning

intervention to document any progress made while those students only received whole class

instruction. After the eight-week intervention period, the AIMSweb TEN will be readministered

to those students whom had received Tier 2 early numeracy intervention. The assessment will

also be given again to a sample of Kindergarten students who did not receive intervention to

begin analyzing and comparing the progress of the intervention students versus those not in

intervention. This method of data collection will be used to help answer both research questions

in this case study.

Data Analysis Methods

For the purpose of this case study, two methods of data analysis will be utilized. When

looking at the qualitative data collected (i.e., observations, field notes, and documents) the

constant comparative data analysis method will be used. For the quantitative data (i.e. AIMSweb

TEN), an independent sample t-test will be used. Both methods of analysis will enable to

researcher to answer both research questions and draw conclusions from the case study.

Constant Comparative Data Analysis

In order to generate themes from the qualitative data collected, the constant comparative

method will be the process used to look at the data over time and constantly compare the data to

determine what it is indicating as to the effect of early numeracy intervention. Creswell (2012)


stated that this method “eliminates redundancy and develops evidence for categories” (p. 434).

Creswell also stated that the constant comparative method will allow the researcher to develop

categories of information from specific to broad which will allow the researcher to generate

themes and draw conclusions. When looking at all of the observations and documents collected,

the researcher will aim to look for evidence of growth in early numeracy. Evidence might

include increased ability to subitize, improved oral counting, increased understanding of the five

frame and base ten system, or the ability to identify numbers within ten. Since the data will be in

the form of observations, intervention logs, and student work samples, the constant comparative

method will allow for a systematic approach of data analysis. Ruona (2005) described that

analyzing qualitative data requires patterns to be discovered through immersion in your data. The

constant comparative method will allow the researcher to begin looking through the data from

the first piece collected to begin identifying emerging trends, as Ruona suggested.

In this case study, student work samples will be collected and observations will be

completed daily electronically via an intervention log. The researcher will use the process Glaser

(1992) suggested when asking questions of the data. The researcher will constantly look through

the observations and documents asking what the data is revealing about student learning in

regards to early numeracy. By critically and constantly looking at the data, the researcher will be

able to find evidence of early numeracy learning and growth that is frequently present within

student work samples and observation. The constant comparative method will allow for the

researcher to notice patterns and connections between observations and student work. The

researcher will be able to group data into relevant categories that seek to answer the impact of

early numeracy intervention among kindergarten students. The process of comparing each piece

of new data to past pieces of data will help the researcher make decisions during the research,


develop additional questions and instructional focuses, reflect on observations, and understand

what trends are emerging in the data (Ruona, 2005).

The constant comparative method will be operationalized through the process of

continuously looking at the data, comparing information, and coding the data into specific

categories for it to be interpreted. The researcher will implement a closed coding system in order

to look specifically for the areas of number sense. Jordan (2007) identified number sense as the

“intuitive knowledge of numbers” (p. 64). Since the research indicated that there is no one

definition of number sense, the researcher will code using many of the components of number

sense such as numeration, base ten proportional reasoning, equality, quantity magnitude, and

form of a number (Faulkner, 2009). Ruona (2005) described the coding process as systematic

and that a coding system should evolve based on the data collected. Immersing in the data will

allow the process of coding to reveal specific categories and trends from the data of which the

researcher will be able to derive meaning from.

Independent Sample T-Test

As part of this case study, the students involved in the early numeracy intervention will

complete the AIMSweb TEN at the beginning, middle, and conclusion of this study. In order to

address the second research question on determining if there’s a difference in early numeracy

progress between a group of students receiving intervention and a group of students not receiving

intervention, an independent sample t-test will be used to analyze data from the AIMSweb TEN

assessments. An independent t-test will allow for the researcher to make comparisons between

two groups from two different settings. This method will allow the researcher to compare the

means between the groups of students participating in the early numeracy intervention with

students who did not receive intervention. The researcher will be able assess if the differences


between the two independent groups is more or less than anticipated for the general population

of students (Creswell, 2012).

For this method of analysis, the researcher will use the five steps in hypothesis testing as

described by Creswell (2012). The first step is determining the null and alternative hypotheses as

stated below.

𝐻𝐴: There is a positive statistical difference within early numeracy growth between

kindergarten students who received intervention as compared with those students that did

not receive intervention.

𝐻0: There is no statistical difference within early numeracy growth between kindergarten

students who received early numeracy intervention as compared to students who did not

receive intervention.

The level of significance (i.e., alpha) for this analysis will be 0.05 indicating that 5% chance that

this researcher’s data results are due to chance. The independent t-test will be a two-tailed test

and give the researcher a p-value that can be used in determining if we can reject or accept the

null hypothesis. When running the analysis, if the p-value is less than our level of significance of

0.05, then we can reject the null hypothesis and accept the alternative (Lesik, 2010). Lesik also

explained when looking at the data analysis if the p-value is less than our level of significance

this implies the value of our test statistic lies in the rejection region allowing us to reject the null

hypothesis. Creswell (2012) and Lesik (2010) both described that in order for the null hypothesis

to be rejected (i.e., which is what the researcher is aiming to accomplish with this case study)

then the p-value needs to inside of the critical regions on the normal bell curve.

The AIMSweb TEN contains for performance task assessments: oral counting, number

identification, quantity discrimination, and identifying missing numbers within a counting


sequence (Clark & Shinn, 2002). Due to this, there will be four independent sample t-tests run to

accurately analyze each section of the AIMSweb TEN. MS Excel will be used to compute the

statistical analysis and then the means between the growths on each section of the AIMSweb

TEN will be compared between the two groups. The researcher will then be able to determine if

there are any statistical differences between the two groups and draw conclusions as to the

impact of early numeracy intervention.

Reliability and Validity

To ensure the reliability and validity of the data collection methods, several steps will be

taken to make sure the data collected and results produced are of high quality and are

dependable. Validating the results of this study will include member checking, an inter rater

reliability check, and triangulation. Creswell (2012) described the process of validating research

findings as a way to determine the accuracy and credibility of results by using various strategies.

To look at the results achieved through qualitative data collection of observations and

student artifacts, the researcher will implement a member check. A member check is a process

where one or more participants check the researcher’s findings for accuracy in describing what

actually took place (Creswell, 2012). Since the participants of this study will be between the ages

of four to five years old, the researcher will utilize a member check with another mathematics

interventionist that conducts the same early numeracy interventions in a different school within

the same district. Since both the researcher and outside mathematics interventionist have had the

same early numeracy training and education, the outside member check will be able to validate if

the researcher has in fact made appropriate conclusions based upon observations and student

artifacts. The process of member checking will enable to researcher to frequently share

observations, themes, and progress based on student work with another individual to determine if


the conclusions and viewpoints taken from the data collected gave an accurate depiction of what

really happened and supported the results (Krefting, 1991). Running the data through a member

check will decrease the chances of misrepresenting the information acquired throughout the

study.

When looking at the quantitative data collected through the AIMSweb TEN, the

researcher will conduct data analysis using an individual sample t-test using an alpha level of

0.05. This assessment had already been tested for inter-rater reliability when first developed.

According to Clarke and Shinn (2002) the AIMSweb TEN was analyzed through alternate-form

reliabilities, inter-rater reliabilities, and test-retest stabilities. The results found that inter-scorer

reliability for all measures was very high. The score was .99 for Oral Counting (OC), Number

Identification (NI), and Quantity Discrimination (QD) measures and .98 for the Missing Number

(MN) measure. Clarke and Shinn stated that when the inter-scorer reliability is over .90,

educational decisions can be made based on the assessment results. Table 2 from Clarke and

Shinn outlines the results from the reliability measurements assessed.

Table 3.2

Early Numeracy Curriculum Based Measurement Reliability for All Testing Sessions


This table allows the researcher to conclude that the AIMSweb TEN is a reliable and valid

assessment tool as evidenced by the data collected from the inter-scorer, alternate-form, and test-

retest measures.

Triangulation will also be used to enhance to reliability and validity of the data collected.

Throughout the data analysis process, themes will be generated through coding and triangulating

the data will allow the researcher to find evidence that supports each theme and result

constructed (Creswell, 2012). Triangulation pulls together all of the data sources and improves

the accuracy of the information. If the results are accurate and reliable, there will be multiple

pieces of data that support each conclusion. Patton (2002) encouraged the use of triangulation by

stating “triangulation strengthens a study by combining methods. This can mean using several

kinds of methods or data, including using both quantitative and qualitative approaches” (p. 247).

Both Patton (2002) and Creswell (2012) described triangulation as a way to improve the

accuracy of a study because the information is drawn from multiple sources. This single case

study should demonstrate the improvement of a child’s early numeracy understanding through

data collected through both qualitative and quantitative methods such as observations, student

documents and artifacts, and the AIMSweb TEN.

Summary

To summarize, the purpose of this single case study will be to examine and explore the

impact early numeracy intervention has on a small group of kindergarten students. Their results

will be compared to a random sample of their kindergarten peers whom did not receive early

numeracy intervention to determine if there was a significant difference between the groups.

Multiple methods of data collection will be implemented including observations, student work

and documents, along with results from an early numeracy assessment called the AIMSweb


TEN. Data will be analyzed through the constant comparative method and through MS Excel by

running an independent sample t-test. The researcher aims to gain a clearer understanding of the

impact early numeracy intervention can have on students showing a gap in their number sense

understanding at the start of kindergarten.


Chapter 4: Results and Discussion

Overview of the Case Study

To begin this single case study, 54 Kindergarten students were administered the

AIMSweb Test of Early Numeracy (TEN) to determine a common ground for their basic

understanding of early numeracy skills. The AIMSweb TEN has four sub-sections of the

assessment: oral counting, number identification, quantity discrimination, and missing number

(see Appendix A). The oral counting section assesses how many numbers from one to one

hundred a student can count. The number identification section assesses how many numbers

ranging one to ten a student can identify. The quantity discrimination section assesses if a child

can determine which quantity is more or less than the other given numbers ranging one to 10.

The missing number section assesses whether a student can fill in the missing number when

given a counting sequence within 10 (Clark & Shinn, 2002).

After the AIMSweb TEN was administered, this researcher looked to identify four to six

students who fell in the below basic proficiency level, as determined by the researcher’s

guidelines set by district this case study took place in. The proficiency standard is achieved when

a student scores a 75% or above on each of the four early numeracy subtests. Students who fell

below that proficiency range in more than two of the sub tests qualified to receive mathematics

intervention. Based off of the data, five students were identified as needing early numeracy

intervention. The early numeracy intervention was delivered to students throughout a six week

intervention cycle, of which those five students received 40 minutes of intervention three days

per week for a total of 120 minutes of Tier 2 intervention weekly. The nature of this study was to

deliver intervention to kindergarten students as early as possible as a key component of effective


intervention is diagnosing a student early and providing tiered intervention (McQuillan &

Connecticut State Department of Education, 2008). The five students whom were identified

during the first week of the school year received four weeks of Tier 1 instruction in their regular

education classrooms prior to beginning Tier 2 intervention in numeracy.

The five students who demonstrated a lack of early numeracy skills had difficulty with

each of the early numeracy tasks on the AIMSweb TEN. Four out of five of the students were

unable to rotely count to ten or demonstrate a solid understanding in accurately counting objects

one by one. Burns (2007) discussed that early counting and one to one correspondence as being

critical to a students’ number sense. Throughout the six-week intervention cycle, these students

participated in hands-on learning activities and research based strategy instruction to improve

their basic numeracy skills. Students worked on slowly building their early numeracy

understandings through counting, subitizing activities, and using games with dice. Opportunities

to play and manipulate numbers, and offering students the opportunity to model mathematics

will help create a stronger number sense (Ginsberg et al., 2005).

Their progress was monitored through student work samples, observations noted on the

intervention log (see Appendix B), and through their AIMSweb TEN data. Both qualitative and

quantitative data was analyzed to answer the research questions in this case study. In order to

gain a larger perspective on the growth of the intervention students, a control group was also

given a second AIMSweb TEN at the end of the intervention cycle in order for their data sets to

be compared. This group was a random sample chosen by the three kindergarten teachers and did

not receive any Tier 2 numeracy support. A statistical analysis through an independent sample t-

test was run to compare the means of both the intervention group and the non-intervention group.


The results from the early numeracy intervention on five kindergarten students will be discussed

in the following sections in relation to the two research questions presented in this case study.

What Is The Impact Of Early Numeracy Intervention On Kindergarten Students?

The impact of early numeracy intervention on kindergarten students is a challenging

question to answer. This case study showed that all of the five intervention students made

significant progress as evidenced by the AIMSweb TEN from their pre-test to post-test. Table

4.1 shows the total scores of the four subsections of the AIMSweb TEN from each students pre-

test, mid-test, and post-test. The mid-test was given to determine the amount of growth that

occurred for each student after four weeks of Tier 1 instruction without intervention.

Table 4.1

AIMSweb TEN Total Score Data

Table 4.1 clearly showed that each of the five students made significant growth from their pre-

test to their post-test. The average growth points from pre-test to the post-test were 49.6 points

after receiving four weeks of Tier 1 instruction and six weeks of Tier 2 intervention. The average

growth from the pre-test to the mid-test of which students were not receiving tiered intervention

was only 8.2 growth points. Table 4.2 describes the standard growth needed for students to meet

the goal proficiency band in each sub-test of the AIMSweb TEN.


Table 4.2

Growth Norms for AIMSweb TEN

GOAL

(Jan -

Sept)/18wks

(growth x

8wk)+ Sept. GOAL (May - Jan)/18wks

(growth x

7wks)+ Jan GOAL

OCM Sept. growth per week. NOV. 1st Jan. growth per week. March 1st May

K 39 1.22 49 61 1.33 70 85

NIM

K 32 1.06 40 51 0.28 53 56

QDM

K 11 0.56 15 21 0.39 24 28

MNM

K 5 0.39 8 12 0.50 16 21

As you can see from Table 4.2, the five students were making adequate amounts of growth in

between the pre and mid-test, however, their starting scores on each of the sub-sections of the

AIMSweb TEN were well below the starting goal range. This allowed the researcher to conclude

that even with Tier 1 mathematics instruction, the students in need of early numeracy

intervention did not make enough growth to end up inside of the goal proficiency band by

November 1st of their school year. This meant they required early numeracy intervention.

Table 4.1 also identified the difference between the growth students made after receiving

tiered intervention as compared to the growth they made after regular Tier 1 instruction is a clear

indicator that the early numeracy intervention had a positive impact on these students. Three out

of five students in fact more than doubled their total scores from the pre to post test. Student C

and Student D also made positive growth, but their growth scores were not as significant when


looking at the other three students’ growth. This data offered the researcher great insight into the

growth track these students might continue to take as they progress in Kindergarten.

To dig deeper into answering the impact that early numeracy intervention had on

kindergarten students, their data from the AIMSweb TEN was also broken down by each of the

four sub sections to identify areas of strength, weakness, and positive growth. Table 4.3

accurately depicts their data on the four sub sections on the assessment.

Table 4.3

AIMSweb TEN Sub Section Data

Table 4.3 showed that in each of the four sub sections of the AIMSweb TEN, each student made

growth from the pre-test, mid-test, to the post-test. This data aided in determining that early

numeracy intervention does positively impact a students’ ability to grow their numeracy skills.

The most significant areas of growth were in the oral counting and number identification

assessments. The data from Table 4.3 also suggested that these students, while making growth,


needed more support in the areas of quantity discrimination and identifying the missing number

from a counting sequence. Student work samples, performance, and observation notes also

support this notion of two sub sections still needing more support.

A child’s ability to be able to subitize quantities is a key foundational piece in their

number sense (Burns, 2007; Penner-Wilger et al., 2007; Van de Walle, 2013). During the work

with these five kindergarten students, none of them were able to subitize past the number three.

In fact, three out of five students could only see the numbers one and two without physically

counting the set. The literature reviewed stated that subitizing was a critical component of a

student’s number sense foundation (Clements, 1999). It was this information that led the

researcher to design targeted interventions where students were working on multiple components

of number sense including the notion of subitizing. Without it, it makes comparing relationships

within numbers as more than, less than, and equal to more difficult. Ginsberg et al. (2005) agreed

with Van de Walle’s (2013) suggestions on the fact that activities to enhance a child’s number

sense must include many hands-on learning experiences. Throughout the six weeks of targeted

intervention, the students were engaged in hands-on learning that aimed to touch upon subitizing,

counting, number identification, number writing, and cardinality.


Figure 4.1. Observation Notes on Intervention Log

In Figure 4.1, the researcher’s observations noted on the daily intervention logs (see Appendix

B) indicate that each student progressed to being able to count rotely 1-20, with three students

being able to count 1-30 without needing prompting. The researcher noted each week that each

of the five students had improved their counting from the previous week citing evidence such as,

“Counted 1-29 before getting stuck; improved” and “counted 1-19 before prompting decades.”

Counting was noted as improving more than five times on the intervention logs. The

observations also show evidence of continued improvement during each week of the intervention

cycle with each student able to count more numbers than the prior week. Figure 4.1 showed

examples for the intervention log that support positive growth. Table 4.2 also supported that each

student improved their counting, for example Student A was only able to count to ten at the start

of the school year and after receiving early numeracy intervention was able to count to 84.

In terms of number identification, Student D was unable to identify any number one

through ten on both the pre and mid-test but was able to identify numbers one through four at the

end of the six-week intervention cycle. The observations made by the researcher show that


Student D was able to identify numbers one and two during week three and was able to identify

the number four inconsistently during week six. Number identification was not an instructional

concept until week three as students had to spend time counting, making matching sets, and

learning how to count objects in a set of five or less by touching one object and saying only one

number. During the first three weeks of the intervention cycle, it was noted that the majority of

students had inaccurate one to one correspondence within the number ten. This was evidenced by

the researcher stating observations such as, “one-to-one is not accurate 100% of the time within

5, concerned with one-to-one and ability to create matching sets, inaccurate counting when

making matching set, and not accurate making a matching set within 5.” All of those

observations were all noted on multiple occasions for each student. The AIMSweb TEN data and

the observation notes all supported these students needing explicit and direct instruction on

counting, counting patterns, one to one correspondence, and their ability to make a matching set

within five prior to addressing any other areas of early numeracy.

Students spent time improving with an activity titled “Grab and Count” that focused on

their one-to-one correspondence and ability to make a matching quantity set (see Appendix C).

This work sample from three students in Figure 4.2 demonstrated the students’ ability to

accurately count and make a matching set of the objects they had just counted.


Figure 4.2. Grab and Count Work Samples

This activity was completed at the end of the third intervention week and all students were able

to accurately count and create a matching set for their numbers within eight as measured by this

activity. This data supported that students had made positive growth in their ability to accurately

count indicating improved abilities in their one to one correspondence when given the task to

create a matching set to their physical objects. Students that were had weak one to one

correspondence within five as evidenced by the observation notes were able to improve by the

time they completed the “Grab and Count” activity. This activity also encouraged number

writing which was a weakness for all five intervention students.

During the first week of the intervention, all students performed a number writing task

with numbers one through twenty (see Appendix D). They also performed the same writing task

during the sixth week of the intervention and you’ll be able to see in their work that growth was


made in their abilities to write numbers. Each student showed positive growth on being able to

write their numbers with more accuracy and precision. Within three weeks all of the five students

were successful at this task, which they were not successful with during week one of the

intervention cycle. This improved ability to count a set of objects accurately, have accurate one-

to-one correspondence and be able to make a matching set also supported the positive impact

early numeracy intervention had on these five students.

In summary, the AIMSweb TEN data and the observation notes along with student work

samples all supported the notion that early numeracy intervention has a highly positive impact on

kindergarten learning. Each of the five students made significant growth between their pre-test

and posttest on each of the AIMSweb sub-tests. The researcher noted many observations where

each of the five students had increased their rote counting skills and improved their

understanding of number within ten. These are the foundational skills of developing a strong

number sense foundation as children cannot learn the numerosity of a set of objects before

learning how to mentally tag each object with a number. Gelman and Gallistel’s (1986)

cardinality principle stated that a building block of number sense if students being able to count a

set of objects and understand that the last number they count is the entire quantity of the set.

Since the students demonstrated poor performance in the area of counting and one-to-one

correspondence at the start of this case study, the improvement indicated in this results section

within their quantitative and qualitative data all suggested that these five students had begun

strengthening their early number sense skills.


Does Early Numeracy Tier 2 Intervention Have A Positive Effect On The Achievement

Gap Between Intervention Students As Compared To Their Peers Not Receiving

Mathematics Intervention In Kindergarten?

In order to address research question 2a (i.e., Does early numeracy Tier 2 intervention

have a positive effect on the achievement gap between intervention students as compared to their

peers not receiving mathematics intervention in kindergarten?), the AIMSweb TEN was

administered to the research group receiving intervention and to a random sample control group

of which students did not receive early numeracy intervention. The data was analyzed through an

independent sample t-test to compare the differences of means on the post-test in Table 4.4.

Table 4.4

AIMSweb TEN Post-Test Comparison

Group N M SD

Intervention 5 80.6 40.65

Control 5 83.6 31.48

The students in the control group had a higher mean score by three points as compared the

intervention group. When running the independent sample t-test, the confidence interval was set

at 0.95 and the alpha value was set at 0.05. The difference was determined to not be statistically

significant with t=0.1305 and p=0.8994. Since the p-value was greater than the alpha of 0.05,

the researcher cannot reject the null hypothesis because the difference between the means of the

two groups is not statistically significant. To answer the second research question, this analysis

showed that there may not have been a statistical difference between the two post-test means,


however, it did show that the intervention group was making progress towards closing the

achievement gap as evidenced by both groups means being within three points of each other.

To further address this research question, the total scores and means were compared

between both group’s pre-test scores. Table 4.5 shows the data from the AIMSweb TEN pre-test

for the group receiving intervention and the control group.

Table 4.5

AIMSweb TEN Pre-Test Comparison

Group N M SD

Intervention 5 15.94 6.72

Control 5 31 56

The difference in means between the intervention and control group on their pre-tests was a clear

indicator of an achievement gap. The control group had scores that gave them a mean that was

almost double of the intervention group’s mean. When running the independent sample t-test, the

confidence interval was 0.95 with an alpha of 0.05. The difference was determined to be

statistically significant with t=3.2577 and p=0.0116. Since the p-value was less than alpha, the

researcher can reject the null hypothesis. More importantly, to address the second research

question, the data from Tables 4.3 and 4.4 indicated that there was a statistical difference

between the means of the two groups at the start of the school year, but that difference was no

longer statistically significant after the intervention group received six weeks of early numeracy

Tier 2 intervention. This answers research question two in justifying that early numeracy

intervention had a positive impact on achievement gap between students receiving intervention

and their peers whom did not.


In summary, each of the five students made significant growth between their pre-test and

post-test, closing the gap between their total mean scores as compared to their non-intervention

peers from a beginning gap of 15.06 points to 3.6 points after only six weeks of early numeracy

intervention. The qualitative data demonstrated improved counting abilities as well as a stronger

one-to-one correspondence when counting within ten. The data presented here also suggested

that the intervention students had in fact almost closed the achievement gap between them and

their non-intervention peers. The goal of a Tier 2 intervention plan is to help the learners begin

achieving at the same standards as their peers. However, the trend the research showed was that

by the time students were receiving Tier 2 mathematics intervention, their gaps in understanding

were too large to be filled in such a short time frame when students present with number sense

deficits in later elementary grades (Bryant et al., 2008). The results shared here show that the gap

between the intervention students and non-intervention students became considerably smaller

after only six weeks of Tier 2 intervention. Clements and Sarama (2004) stated that mathematics

intervention needs to begin in the early grades in order to improve on their number sense abilities

before it is too late and the academic gap widens. Clearly, the early numeracy intervention had a

positive effect on enabling these five students to catch up with their non-intervention peers as

evidenced throughout this results section.

These results prove the effect that providing targeted intervention as early as possibly can

have on students and their mathematics abilities. Educators need to utilize early numeracy

assessments that focus on what Jordan (2006) identified as the common core areas of

misconceptions: counting sequence, quantity discrimination, number transformation, estimation,

and extending number patterns. According to Jordan, these were skills that preceded a child’s

ability to conceptually understand mathematics in later years. This is what sparked the researcher


to use the AIMSweb TEN in an effort to identify learning gaps within the kindergarten

population before the gap widened. The data speaks to Jordan’s notion that a child cannot be

successful in later mathematics without a strong foundation. In just six weeks, five students made

tremendous growth as compared to their non-intervention peers, growth they may not have made

without the early numeracy intervention.

Is There Any Difference In Progress Within The Two Groups?

Research question 2b (i.e., Is there any difference in progress within the two groups?)

was perhaps the most challenging to answer based on this single case study. The researcher had

no intervention with the students whom did not receive early numeracy intervention on a daily

basis. While the researcher had access to the scope and sequence of the Tier 1 mathematics

curriculum, there was no qualitative work samples or observations to accurately compare

because the researcher did not provide the Tier 1 instruction to the students outside of the early

numeracy intervention group. With that said, the AIMSweb TEN data between the two groups

does suggest a difference within their progress.

In Table 4.6, the pre-test and post-test AIMSweb scores are compared between the two

groups. This offered insight into the amount of progress each student within each group had

made throughout the first ten weeks of their kindergarten school year.


Table 4.6

Comparison of AIMSweb TEN Total Scores

Student Total Score Pre Total Score Post GROWTH

A 23 142 119

B 9 58 49

C 43 56 13

D 31 45 14

E 49 102 53

AVG 31 80.6 49.6

Student Total Score Pre Total Score Post Growth

A1 51 61 10

B1 65 126 61

C1 49 48 -1

D1 55 103 48

E1 61 80 19

AVG 56.2 83.6 27.4

Table 4.6 demonstrated the differences between the total AIMSweb TEN scores within the two

groups. Students A through E were among the early numeracy intervention group. Their average

growth after ten weeks of Tier 1 instruction on top of six weeks of Tier 2 mathematics

intervention was 49.6 points. Students A1 through E1 were in the control group and had an

average growth increase of 27.4 points. The intervention group made 22.2 points more growth in

the same amount of time as compared to their non-intervention peers.


These results indicated that the Tier 2 intervention enabled these students to make more

progress within the same time period over their non-intervention peers. It should be noted

however, that the starting scores for the non-intervention group had an average of 56.2 points,

whereas the intervention group was starting with an average of 31 points. In the overall theme of

developing number sense, the intervention students started at a much lower range in their

understanding of basic number concepts and had much more growth to make in order to meet the

goal proficiency bands on each of the AIMSweb subtests. The five non-intervention students had

already begun the year within the appropriate profiency ranges; the intervention students did not.

McQuillan & CSDE (2008) discussed that the overall goal of a Tier 2 intervention plan is

to help the learners begin achieving at the same standards as their peers. In order for this to be

possible, students must often making more than one year’s worth of growth in one year’s time.

The data in Table 4.6 suggested that the intervention students were on their way to making more

than one year’s worth of growth based on the fact that their average growth score was almost

double when compared to their non-intervention peers. To address the research question, this

does not adequately answer if there is a difference in progress between the two groups, only that

the data proves both groups are making progress with the Tier 2 intervention students making

more growth. This is, in fact, the goal of RTI Tier 2 intervention and was exactly what the

researcher wanted to observe.

In summary, more qualitative and quantitative data would be needed as both groups of

students continued throughout their kindergarten year. In order to fully address this research

question, the students’ data would need to be compared at the end of their kindergarten school

year to completely determine if the progress was different between students only receiving Tier 1


mathematics instruction and those whom have received Tier 1 and Tier 2 mathematics

instruction.

Chapter Summary

This chapter discussed the results among qualitative and quantitative data collected

throughout this single case study. The researcher wanted to explore the impact of early numeracy

and if Tier 2 early numeracy intervention would have a positive effect on the achievement gap

among kindergarten students. A critical conclusion from a variety of intervention studies was that

students needed to have their understanding of number addressed at earlier ages (Browder et al.,

2012; Bryant et al., 2008; Douglass & Horstman, 2011; Fuchs, 2008; Yung & Reifel, 2010). This

case study focused on kindergarten students and the results indicated that addressing these

students’ deficits in early numeracy had a positive impact on their overall number sense. In only

six weeks, the gap between the total scores on the AIMSweb TEN between both groups of

students was minimal. Early intervention focused on number sense proved to benefit all of the

five students in this case study.

To conclude, this single case study was successful at increasing the understanding of five

students’ early numeracy abilities. Each student improved on all of the four sub-tests of the

AIMSweb TEN and their total average growth scores were within three points of the control

group of students whom did not receive intervention. This suggested the achievement gap was

closing and the Tier 2 early intervention was successful. The students will, however, continue to

receive Tier 2 mathematics intervention until their data can be tracked and compared over a

longer period of time. These five students were seen consistently three days a week over a six-

week period and the observations made on each student were kept with fidelity after each

intervention session.


Chapter 5: Conclusion

Summary of Single Case Study

The purpose of this single case study was to explore the impact earl numeracy

intervention would have on kindergarten students. The literature reviewed for this research

indicated a clear need for further study to be done on early numeracy intervention beginning with

identifying the gaps in a students’ number sense and designed targeted interventions to improve

their sense of number within a Tier 2 Response to Intervention (RTI) model. This researcher

sought to determine if a gap in a child’s mathematical understanding was addressed at an earlier

age, will that student be able to close the gap with their peers and return to Tier 1 instruction. The

specific research questions in this single case study were:



gap between intervention students as compared to their peers not receiving mathematics



Within a six week intervention period, this case study provided valuable information that

addressed each of those research questions.

The quantitative data from the AIMSweb Test of Early Numeracy (TEN) and the

qualitative data from the researcher’s observation and student work samples presented in the

results section all indicated that each of the five students receiving Tier 2 mathematics

intervention made positive growth in their early numeracy understandings. Each of the five

students improved their rote counting, understanding of cardinality, one-to-one correspondence,


number identification, quantity discrimination and their counting pattern sequences. The

literature reviewed suggested many different formal definitions of number sense, but all included

counting, cardinality, number identification, and being able to understand quantities of numbers

as part of a child’s number sense. A student cannot begin to understand more challenging

mathematical concepts such as place value or computation prior to having a foundation in

number sense (Griffin, 2004, Van de Walle, 2013). Tier 2 early numeracy intervention was

needed to help support the five learners in this case study.

Each research question was answered through analyzing the qualitative and quantitative

data to identify improvement in each students’ number sense ability. As discussed in Chapter 4,

each student made significant growth as measured on the AIMSweb TEN, observation notes, and

student work samples. Each student was able to extend the counting sequence to at least twenty,

which three out of five students were unable to do at the start of this study. In fact, those three

students were unable to count accurately even within five at the start of their kindergarten school

year. When compared to their peers, whom many of could count to at least thirty, this was

already a significant gap in just one area of number sense. The AIMSweb TEN data presented in

Chapter 4 clearly demonstrated that each student made positive growth when assessed in four

sub-sections of early numeracy. The observer’s notes also indicated improvements in each

students’ understanding of cardinality and being able to accurate count sets of objects within ten.

While to someone outside of the educational realm, this may seem to be such small gains, in the

world of mathematical understanding, a child will have a difficult time achieving success in later

concepts if they cannot count of understand exactly what a numeral represented.

This single case study proved that early numeracy intervention could help a student

improve their overall numeracy understanding, which would have been more challenging to do


in later years. Through targeted intervention and evidence based data making, the RTI Tier 2

intervention was successful. Given that this case study only analyzed data after six weeks of Tier

2 intervention, the long-term effects of early numeracy intervention cannot be commented on.

However, the amount of progress each student made within six weeks was encouraging to the

researcher that these students may be able to fully close the achievement gap and return to only

receiving Tier 1 mathematics intervention at some point in elementary school.

It was beneficial for these students to receive Tier 2 early numeracy intervention after

only four weeks of Tier 1 instruction alone because Lembke, Hampton, and Beyers (2012)

suggested that in order for the systematic instruction of Tier 2 intervention to be successful,

educators must build a students’ mathematical understanding in a logical order. It was more

efficient to provide those five students with early numeracy intervention while they are still in

the early primary grades at ages four and five. The gap in their understanding of number would

have been much more challenging to begin closing at a later age when the mathematical

expectations were substantially increased.

This single case study proved that when educators can intervene as early as possible,

students can make more than expected progress and be able to start closing the achievement gap

with their peers. This study also proved the power that hands-on targeted instruction has on our

youngest learners. Each intervention lesson was determined based on evidence from data and

student work. This ability to target instruction to meet the independent needs of kindergarten

students enabled these five students to learn at a rate faster than their peers and encouraged

growth in many of the critical areas of number sense. Without a strong numeracy foundation,

these students would absolutely have difficulty in later, more challenging, mathematical

concepts. It is the hope that this early numeracy intervention will ensure the positive growth of


these five kindergarten students throughout their public education. Early numeracy intervention

in kindergarten had an immensely positive impact on these five students and their understanding

of number.

Implications for Practice

Throughout this researcher and case study, the impact that early numeracy intervention

had on five students should not be taken lightly. For all educators, this study helped to prove how

the achievement gap is easier to close within mathematics if students receive intervention as

early as possible. In the many studies reviewed on mathematics intervention, it was determined

that often a child was struggling in mathematics due to gaps in their number sense foundation.

These gaps are difficult to address when the child progresses in elementary school and needs to

learn about higher level concepts such as fractions if they cannot understand whole number

counting sequences, or need to count the dots on dice before knowing how many are there. While

the five students whom received the intervention will all continue to in this Tier 2 environment,

they proved that the additional instruction benefited their overall number sense ability. What this

means for educators, is that they need to intervene when students are demonstrating difficulties

as early as possible. There had been a misconception in the school system that this case study

was performed in that we needed to give students time in Tier 1 for as long as possible. If we

could give them effective Tier 1 instruction for a couple years, they would catch up. This

researcher believed that intervention focused on early numeracy could not wait. Much of the

research reviewed discussed mathematics intervention beginning in grade three. Starting Tier 2

or Tier 3 intervention with students three years after they’ve entered kindergarten, where they

already had gaps in their number sense, is too late for the RTI practice to work. The skills that


these five students lacked are all grounded in number sense, many of which are developed prior

to kindergarten such as subitizing and counting to ten. This study would not have been effective

in closing some of the achievement gap if educators had waited until later primary grades to

provide targeted early numeracy instruction. Educators in public school systems must begin to

address numeracy deficits in kindergarten.

Another implication of practice that was learned was the need for rich learning

experiences for kindergarten students. While Connecticut is implementing the Common Core

State Standards in Mathematics (CCSS-M), these standards must be aligned to effective teaching

practices such as hands-on learning experiences. The five intervention students participated in

targeted hands-on learning surrounding early numeracy concepts three days a week. Griffin

(2004) supports the notion that educators must provide students rich activities for making

connections, exploring and discussing concepts, and ensuring an appropriate sequence of

concepts.

The intervention group made limited progress as evidenced on the AIMSweb TEN after

four weeks of only Tier 1 mathematics instruction. This can lead to the conclusion that the Tier 1

learning environment may not have contained appropriate hands-on learning activities that

allowed the students to explore early numeracy concepts. Given the limited skills of the five

kindergarten students, it was assumed that their Tier 1 instruction was not targeted to where they

were as learners. Cady and Hopkins (2007) discussed in their researcher the need for students to

work with manipulatives where they build an understanding of the number five prior to leaping

to understanding ten. Since the intervention group had trouble with their one-to-one

correspondence within five, they were simply not ready to address sets of ten which is a standard

in the kindergarten CCSS-M. What this means for educational practice is that even though


kindergarten teachers in CT need to address the CCSS-M, they cannot forget that they have to

teach a child at their own instructional level.

Suggestions for Further Research

In the area of understanding the impact early numeracy intervention has on students, a

study would need to be conducted with these five students as they progress in elementary school.

While each of the students in the intervention group made positive growth in their number sense

ability, six weeks of targeted Tier 2 intervention is not long enough to study the lasting effects

this early intervention had. The need for this study was evidence within the literature reviewed;

as many of the cases studied suggested that while Tier 2 intervention helped a child make

progress in their mathematics ability, it was not yet helpful in closing the achievement gap

between intervention students and their non-intervention peers because the interventions had not

been provided early enough. For example, Bryant et al. (2008) made a case for Tier 2

mathematics intervention to include number sense concepts because it is such a large predictor

for future math difficulties. Future research studies will need to be done as to determine the true

impact early intervention has on students throughout their educational journey.

There are several questions that the researcher would still like to answer as a result of this

single case study. The first is will the intervention students continue their positive trend in

student achievement and be able to perform and understand mathematics as well as their non-

intervention peers? As with any RTI approach, there is not a clear “end game.” There is no clear

outline from the Connecticut State Department of Education as to how long to keep students in

Tier 2 intervention if they are making continued progress. Further research will be needed as to

how to determine when a student is ready to exit Tier 2 early numeracy intervention and how to

continue to monitor their progress within the Tier 1 mathematics classroom.


Another aspect that needs further research is the impact preschool had on these

kindergarten students. While not discussed, three out of the five students did not attend a

formalized preschool program. If they had attended preschool, would they have needed early

numeracy intervention? Since the adoption of the CCSS-M, kindergarten students are held to a

set of standards, some of which mirror what used to be taught in first grade prior to the adoption

of the CCSS-M. Further investigation is needed to determine how preschool impacts a

kindergarten students’ number sense development.

Limitations

There are three main limitations that this research study was affected by. One is that this

single case study only contained five kindergarten students. The students identified as needing

mathematics intervention were then compared to the same size sample group to their peers not

receiving intervention. The results in the case study only looked at a total of twelve students,

which in comparison to other research studies, is a small sample size when discussing the results.

It was difficult to generalize the impact of early numeracy intervention when working within

such a small sample.

The second limitation was that the interventionist who provided research-based

instruction was not be a part of the control group’s instruction or targeted instructional plans. It

was assumed that these students are performing moderately well in Tier 1 of their classrooms.

This conclusion was drawn by using their AIMSweb TEN data from the start of the academic

school year. The control group had to have scores that fell into the goal proficiency band on unit

assessments and the AIMSweb TEN. However, the interventionist was not able to monitor the

type of instruction that was being implemented in Tier 1. For the purpose of this study though, it

was more important that the growth of student learning was looked at within the intervention


group receiving extra numeracy support as compared to the group of students in Tier 1 not

receiving extra support outside of Tier 1 mathematics instruction.

The last limitation was the aspect of time. For the purpose of this case study, the

participants were observed over a ten-week period. The first four weeks aimed to see how much

achievement growth was made while only receiving Tier 1 instruction, and the later six weeks

included the students receiving the intervention support. In order to truly make an argument that

early numeracy instruction was effective at closing the achievement gap in mathematics, these

kindergarten students would need to be monitored throughout their elementary school years to

determine if they can remain proficient in mathematics without needing further intervention after

having early numeracy intervention. A longitudal study was simply not possible given the time

frame of this research study.


References

Adler, P. A., & Adler, P. (1994). Observational techniques. Handbook of qualitative research (1st

ed.). (pp. 377-392), Thousand Oaks, CA: Sage.

Altheide, D. L., & Schneider, C. J. (2012). Qualitative media analysis (Vol. 38). Thousand Oaks,

CA: Sage.

Battista, M. T. (2012). Cognitive based assessment and teaching. Portsmouth, NH: Heinemann.

Berch, D. B. (2005). Making sense of number sense implications for children with mathematical

disabilities. Journal of Learning Disabilities, 38(4), 333-339.

Berch, D. B., & Mazzocco, M. M. (2007). Why is math so hard for some children? In the The

Nature and Origins of Mathematical Learning Difficulties and Disabilities (pp. 369-441).

Baltimore, MD, Paul H. Brookes Pub. Co.

Bouck, E. C., Satsangi, R., Doughty, T., & Courtney, W. T. (2014). Virtual and concrete

manipulatives: A comparison of approaches for solving mathematics problems for

students with autism spectrum disorder. Journal of Autism and Developmental Disorders,

44, 180-193.

Browder, D. M., Jimenez, B. A., Spooner, F., Saunders, A., Hudson, M., & Bethune, K. S.

(2012). Early numeracy instruction for students with moderate and severe developmental

disabilities. Research and Practice for Persons with Severe Disabilities, 37(4), 308-320.

Bryant, D. P., Bryant, B. R., Gersten, R., Scammacca, N., & Chavez, M. M. (2008). Mathematics

intervention for first and second grade students with mathematics difficulties. Remedial

and Special Education, 29(1), 20-32.


Bryant, D., Bryant, B., Gersten, R., Scammacca, N., Funk, C., Winter, A., Pool, C. (2008). The

effects of tier 2 intervention on the mathematics performance of first grade students who

are risk for mathematics difficulties. Learning Disability Quarterly, 29(1), 20-32.

Burns, M. (2007). About teaching mathematis: A K-8 resource (3rd ed.). Sausalito, CA: Math

Solutions.

Cady, J., & Hopkins, T. (2007). What is the value of @ #? Deepening teachers' understanding of

place value. Teaching Children Mathematics, 13(8), 434-438.

Cain, C., & Faulkner, V. (2012). Teaching number sense in the early elementary years. Teaching

Children Mathematics, 18(5), 288-294.

Clarke, B., & Shinn, M. (2002). Test of early numeracy: Administration and scoring of AIMSweb

early numeracy measures. Eden Prairie, MN: EdFormation Inc.

Clarke, B., Smolkowski, K., Baker, S., & Chard, D. J. (2011). The impact of a comprehensive

tier 1 core kindergarten program on the achievement of students at risk in mathematics.

The Elementary School Journal, 111(4), 561-584.

Clements, D. (1999, March). Subitizing: What is it? Why teach it? Teaching Children

Mathematics, 5(7), 399-405.

Clements, D. H., & Sarama, J. (2011). Early childhood mathematics

intervention. Science, 333(6045), 968-970.

Cooper, L. L., & Tomayko, M. C. (2011). Understanding place value. Teaching Children

Mathematics, 17(9), 558-567

Creswell, J. H. (2003). Research design: Qualitative, quantitative, and mixed methods

approaches (2nd ed.). Thousand Oaks, CA: Sage.


Creswell, J. H. (2012). Educational research: Planning, conducting, and evaluating quantitative

and qualitative research (4th ed.). Upper Saddle River, NJ: Pearson Education, Inc.

Crotty, M. (1998). The foundations of social research: Meaning and perspective in the research

process. London: Sage.

Devers, K., & Frankel, R., (2000). Study design in qualitative research—2: Sampling and data

collection strategies. Education for Health, 13(2), 263–271

Douglass, L., & Horstman, A. (Fall 2011). Integrating respsonse to intervention in an inquiry

based math classroom. Ohio Journal of School Mathematics, 64(1) , 23-29.

Faulkner, V. (2009). The components of number sense. Teaching Exceptional Children, 41(5),

24-30.

Fosnot, C. (2010). Models of intervention in mathematics: Reweaving the tapestry. Reston, VA:

The National Council of Teachers of Mathematics, INC.

Fosnot, C., & Dolk, M. (2001). Young mathematicians at work: Construction number sense,

addition, and subtraction. Portsmouth, NH: Heinemann.

Fuchs L. S., Fuchs, D., Powell, S. R., Seethaler, P. M., Cirino, P. T., & Fletcher, J. M. (2008).

Intensive intervention for students with mathematics disabilities: Seven principles of

effective practice. Learning Disability Quarterly, 31(2), 79-92.

Fuchs, D., Mock, D., Morgan, P. L., & Young, C. L. (2003). Responsiveness-to-intervention:

Definitions, evidence, and implications for the learning disabilities construct. Learning

Disabilities Research & Practice, 18(3), 157-171.

Fuson, K., & Briars, D. (1990). Using base ten blocks learning/teaching approach for first and

second grade place value and multidigit addition and subtraction. Journal for Research in

Mathematics Education, 21(3), 180-206.


Gelman, R., & Gallistel, C. R. (1986). The child's understanding of number. Harvard University

Press.

Ginsburg, A., Cooke, G., Leinwand, S., Noell, J., & Pollock, E. (2005). Reassessing U.S.

international mathematics performance: New findings from the TIMSS and PISA.

Washington, DC: American Institutes for Research.

Gresham, F. M. (2009). Using response to intervention for identification of specific learning

disabilities. School Psychology, 9, 205-220.

Griffin, S. (2004). Teaching number sense. Educational Leadership, 61(5), 39-42.

Hodges, T., Rose, T., & Hicks, A. (2012). Interviews as RTI tools. Teaching Children

Mathematics, 20(8), 30-36.

Hynes, M., (1986). Selection criteria. Arithmetic Teacher, 33(6), 11-13.

Jordan, N. (2007). The need for number sense. Educational Leadership, 65(2), 63-66.

Jordan, N., Glutting, J., & Ramineni, C. (2009). The importance of number sense to mathematics

achievement in first and third grades. Learning and Individual Difference, 20, 82-88.

Jung, H. Y., & Reifel, S. (2010). Promoting children’s communication: A kindergarten teacher’s

conception and practice of effective mathematics instruction. Journal of Research in

Childhood Education, 25, 194-210.

Kearns, D. M., & Fuchs, D. (2013). Does cognitively focused instricion improve the academic

performance of low-achieving students? Exceptional Children, 79(3), 263-290.

Koellner, K., Colsman, M., & Risley, R. (2011). Multidimensional assessment: Guiding response

to intervention in mathematics. Teaching Exceptional Children, 44(2), 2-14.


Lampert, M. (1990). When the problem is not the question and the solution is not the answer:

Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29-

63.

Lembke, E. S., Hampton, D., & Beyers, S. J. (2012). Response to intervention in mathematics:

Critical elements. Psychology in the Schools, 49(3), 257-272.

Lesik, S. (2010). Applied statistical inference with minitab. Boca Raton, FL: Taylor & Francis

Group.

Mathematics, N. C. (2000). Principles and standards for school mathematics. Reston, VA:

National Council for Teacher of Mathematics.

McCallum, B., Zimba, J., & Daro, P. (2010). Common core state standards mathematics.

Washington, D.C: Common Core State Standards Initiative. Retrieved from

http://www.corestandards.org/Math/

McGuire, P., Kinzie, M., & Berch, D. (2012). Develop number sense in pre-K with five frames.

Early Childhood Education, [Missing volume number], 213-222.

McQuillan, M., & Connecticut State Department of Education, C. (2008). Connecticut's

framework for RTI. Hartford, CT: Connecticut State Department of Education.

McQuillan, M., & Connecticut State Department of Education. (2008). Connecticut's framework

for RTI. Connecticut State Department of Education.

Nelson, C. (March 2012). The math box project. Teaching Children Mathematics, [Missing

volume number], 418-425.

Patton, M. Q. (2002). Qualitative evaluation and research methods (3rd ed.). Thousand Oaks,

CA: Sage Publications, Inc.


Penner-Wilger, M., Fast, L., LeFevre, J., Smith-Chant, B. L., Skwarchuk, S., Kamawar, D., &

Bisanz, J. (2007). The foundations of numeracy: Subitizing, finger gnosia, and fine-motor

ability. In Proceedings of the 29th annual cognitive science society (pp. 1385-1390).

Austin, TX: Cognitive Science Society.

Powell, S. R., & Stecker, P. M. (2014). Using data-based individualization to intensify

mathematics intervention for students with disabilities. Teaching Exceptional Children,

46(4), 31-37.

Puchner, L., Taylor, A., O'Donnell, B., & Fick, K. (2005). Teacher learning and mathematics

manipulatives: A collective case study about teacher use of manipulative in elementary

and middle school. School Science and Mathematics, 108(7), 315-324.

Ross, S. H. (1989). Parts, wholes, and place value: A developmental view. Arithmetic

Teacher, 36(6), 47-51.

Ruona, W. E. (2005). Analyzing qualitative data. In R. A. Swanson & E. F. Holdton III (Eds.),

Research in organizations: Foundations and methods of inquiry (pp. 233-263). San

Francisco, CA: Berrett-Koehler Publishers, Inc.

Sandelowski, M. (2000). Focus on research methods-whatever happened to qualitative

description? Research in Nursing and Health, 23(4), 334-340.

Smith, K., & Geller, C. (2004). Essential principles of effective mathematics instruction:

Methods to reach all students. Preventing School Failure, 48(4), 22-29.

Stipek, D., Schoenfled, A., & Gomby, D. (2012, November). Math matters. Educational Week.

Swan, P., & Marshall, L. (2010). Revisiting mathematics manipulative materials. Primary

Mathematics Classroom, 15(2), 14-19.

https://blackboard.sacredheart.edu/webapps/blackboard/execute/content/file?cmd=view&content_id=_1842138_1&course_id=_15991970_1




Van de Walle, J. A., Karp, K., & Bay-Williams, J. (2013). Elementary and middle school

mathematics: Teaching developmentally. Upper Saddle River, NJ: Pearson Inc.

Walle, J. V., & Lovin, L. A. (2005). Teaching student centered mathematics: Grades K-3. Upper

Saddle River, NJ: Pearson Inc.

Wynn, K. (1992). Children's acquisition of the number words and the counting system. Cognitive

Psychology, 24(2), 220-251.

Yin, R. K. (2003). Case study research: Design and methods (3rd ed.). Thousand Oaks, CA:

Sage.

THE IMPACT OF EARLY NUMERACY INTERVENTION Appendix A 83

DATE NAMES LESSON OBJECTIVES COMMENTS

9/28/15 A

B

C

D

E

Rote counting practice 1-30

Counting and Cardinality: Count the number

of blocks in your pile within 10

1:1 Counting Strategy modeled- touch one, say

one, move one

A: Accurate 1:1 within 6

B: Able to count to 7 from memory

C: Accurate 1:1 within 10

D: Refused to count pile of blocks independently

E: Left early for dismissal

9/29/15 A

B

C

D

E

Rote counting practice 1-30

Counting and Cardinality: Count the number

of blocks in your pile within 10

1:1 Counting Strategy modeled- touch one, say

one, move one

A: Accurate 1:1 within 11

B: Accurate 1:1 within 6, able to count to 10

C: Accurate 1:1 within 13, rote counting inaccurate during teen

numbers

D: Accurate 1:1 within 6, rote counted to 10

E: Needs to practice touching one block while saying one

number

10/1/15 A

B

C

D

E

Rote Counting Practice 1-30 on life size number

line

Counting Backwards from 10

1:1 Counting Strategy model within 15

Number Writing Sample

A: Accurate 1:1 within 11, could not ID matching #

B: Could not write any numbers after 3, counted to 30 with

only two errors

C: Accurate 1:1 and number matching

D: Lack of effort and focus, did not count today

E: Accurate 1:1 today, able to find matching number ID with

guidance of counting on from 1

Writing Samples: Able to write 1-3 for all

THE IMPACT OF EARLY NUMERACY INTERVENTION Appendix B 87


10/5/15 A

B

C- ABS

D

E

Rote counting practice 1-50 on the 100 chart

Match my math rack activity within 5.

*Model 1:1 counting, touch one, say one, move one

*gradual practice of 1:1 within 5

A- Counted 1-29 before getting stuck; improved

B- Counted 1-17 indedendently, improved, 1:1 is not

accurate 100% of the time within 5

C- ABS

D- little effort; counted to 15 when pushed, unable to

create matching images on math rack

E- Counted to 30 before getting stuck; improved; 1:1

within 10 is inconsistent

10/6/15 A

B

C

D

E

Rote counting practice 1-50 on the 100 chart

Match my math rack activity within 20

Continue working on 1:1 counting on math rack

within 20

A- Improved 1:1 within 20, counted math rack accurately

B- Not motivated, watching others before moving beads on his

math rack, counting is improving

C- Excellent effort, accurate counting to 20, able to match math

rack within 5

D- Little counting effort, concerned with his 1:1 and ability to

create matching sets

E- Counting improving, 1:1 was accurate within 5 today

10/8/15 A

B

C

D

E

Rote counting 1:1 to 50

Match my math rack: Combinations of 5

Early Subitizing Activity: Using dot plates (1-5)

count how many and make a matching set.

A- Accurate counting through 39, able to make matching set

using counters

B- Rote counted to 30, able to make matching set using

counters

C- Difficulty counting through teen numbers today but

recognizes pattern, able to make matching set using counters

D- No effort counting, able to make matching set using

counters

E- Accurate counting through 30, able to count and make

matching set using counters



10/12/15 A

B

C

D

E

NO SCHOOL

10/13/15 A:

B

C: MOVED

D- ABS

E

Rote counting using 100 chart 1-50, stopped to

prompt at decades

Early subitizing dot plate activity, make a

matching set within 5

Dream Box

A: Inaccurate counting when making matching set, can

subitize within 2, counted 1-29 before prompting

B: counted 1-29 before prompting decades, subitize within 2,

not accurate making a matching set within 5, unsure “how

many” means to count

E: Able to subitize within 3, counted 1-29 before needing

prompting, accurate matching sets within 5

10/15/15 A

B

C

D

E

Rote counting using 100 chart 1-50, stopped to

prompt at decades

Early subitizing dot plate activity, make a

matching set within 5

A: Counted 1-29 before prompting of the decade, able to make

a matching set within 5 today

B: counted 1-29 before prompting decades, subitize within 2,

not accurate making a matching set within 5

D: Counted to 20, inaccurate when making a matching set

E: Able to subitize within 3, counted 1-29 before needing

prompting, accurate matching sets within 5



10/19/15 A

B- absent

C-absent

D

E-absent

Rote counting 1-50 without 100 chart

Subitizing and making equal sets

o Roll Dice

o Count out the same number of cubes as the

number on the die

o Who has more?

A: Counted 1-39 before prompted for 40, can subitize w/in 4

D: Counted 1-14, prompted 15 and 16, continued counting

17,18,19, prompted at 20, counted 21-25 and stopped

Cannot subitize at all after 1

10/20/15 A

B

C

D

E- absent

Rote counting 1-50 without 100 chart

Subitizing and making equal sets

o Roll Dice

o Count out the same number of cubes as the

number on the die

o Who has more?

A: Counted 1-49 before prompted for 50, able to subitize 1, 2,

3, 4, and 6 today

B: slow counting pace, counted 1-19, skipped 20, completely

lost and unable to say next number after 25

C: counted 1-4, skipped 15, 18, and prompted for 20

D: Counted 1-14, skipped 15, 17, 19, and 20…

10/22/15 A

B

C

D

E- absent

Number ID activity with digit Dds through 10.

Find the number after 4. What number comes

before 10? (etc.)

Grab and Count

o Grab handful of cubes

o Count 1:1 How Many?

o Draw a matching picture

o Write the corresponding number

A: Eager, impulsive, accurate counting within 10. Able to

identify numbers only when she counted up from 1.

B: Able to ID numbers 1-5, this is improved! Accurate 1:1

counting and matching picture within 8

D: Unable to identify any numbers after 1, 2, and 3. Successful

counting cubes 1:1 within 8 and drawing matching picture



10/26/15 A

B

C

D

E

Counting video and practice 1-50

Racing Bears game

o Number identification 1, 2, 3

o 1:1 within 10

o Counting on within 10

A: Able to identify numbers 1-3 with numerals only

B: Unable to identify numbers 1-3 without dots to represent

amounts

C: Able to identify numbers 1-3 with numerals only

D: Guessed the number 5 for all numerals, unable to identify

numbers 1-3 even with dot visual

E: Able to identify numbers 1-3 with numerals only

10/27/15 A

B

C

D

E

Counting video and practice 1-50

Racing Bears game


o 1:1 within 10

o Counting on within 10 (a few students

within 20)

10/29/15 A

B

C

D

E

Rote counting 1-50

Number identification 1-5 game

Racing Bears game


o 1:1 within 20

o Counting on within 20

A: counted 1-69 prompted 50, 60, 70, able to ID #’s 1-10

B: counted 1-30, prompted with 31, able to identify numbers 1

and 2

C: counted 1-19 (skipped 16), able to identify #’s 1-4

D: Counted 1-18 (skipped 15), able to identify #’s 1-3

E: Counted 1-28 before needing to be prompted, able ID

numbers 1-10



11/2/15 A

B

C

D

E

Rote counting 1-50 with 100 grid

Number Identification Activities: Circle 1’s, cross

out 2’s, underline 3’s, Triangle 4’s, Erase 5’s

Number writing practice 1-5

Dream Box

Overall, counting is improving but teen numbers are a

struggle for C, B, and D

-E: identified all #s 1-10

-B: identified 1s, and some 2’s, cannot ID 3-5

-D: Unable to do more/less activities on Dream Box

11/3/15 A

B

C

D

E

NO SCHOOL

11/5/15 A

B

C

D

E

Rote counting 1-50 using 100 chart

Number identification activity

Number writing 1-20 sample

1:1 counting activity within 20

A: Counted to 50!! No prompting, identified all numbers 1-5

B: Counted to 30, identified numbers 1 and 2, very difficult

time number writing poor pencil grip, no accurate 1:! After 10

C: Identified numbers 1-5, number writing has improved

within 10, accurate 1:1 within 20

D: number writing improved, 1:1 accurate within 10, identified

numbers 1-4

E: Counted to 50! Identified all numbers 10, accurate 1:1 within

20, improved number writing within 20

11/6/15 A

B

C

D

E

AIMSweb Test of Early Numeracy Ereen

(End of Intervention Cycle 1)


THE IMPACT OF EARLY NUMERACY INTERVENTION Appendix C 93

Student A Pre-Sample

THE IMPACT OF EARLY NUMERACY INTERVENTION Appendix D 97

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Student A Post-Sample


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Student B Pre-Sample


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Student B Post-Sample


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Student C Pre-Sample


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Student C Post-Sample


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Student D Pre-Sample


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Student D Post-Sample


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Student E Pre-Sample


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Student E Post-Sample


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THE IMPACT OF EARLY NUMERACY INTERVENTION Appendix E 106

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THE IMPACT OF EARLY NUMERACY INTERVENTION Appendix F 118